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NEW COMPLETE 
BUSINESS ARITHMETIC 


GEORGE 



: VAN TUYL 


EVANDER CHILDS HIGH SCHOOL, CITY OF NEW YORK 


INSTRUCTOR IN BUSINESS MATHEMATICS IN THE SCHOOL OF COMMERCE, ACCOUNTS 
AND FINANCE, NEW YORK UNIVERSITY, AND IN BUSINESS 
EXTENSION COURSES, COLUMBIA UNIVERSITY 
FORMERLY TEACHER OF BUSINESS ARITHMETIC IN THE PACKARD COMMERCIAL 
SCHOOL AND IN THE HIGH SCHOOL OF COMMERCE, 

CITY OF NEW YORK 





AMERICAN BOOK COMPANY 


NEW YORK CINCINNATI 


CHICAGO 


BOSTON 


ATLANTA 














, sVt 


Copyright, 1911, 1915, by 
GEORGE H. VAN TUYL. 


Copyright, 1924, by 
AMERICAN BOOK COMPANY 
All Rights Reserved 


NEW COMP. BUS. ARITH. 

w. p. 1 


4 it 

<t if » 

< 


Made in U. S. A. 

APR - 3 *24 

©Cl A7 77702 

fn I 




PREFACE 


In the preparation of New Complete Business Arithmetic 
the author has kept in view the same specific objects which 
characterize his Complete Business Arithmetic. Briefly these 
objects are. 

Training that leads to facility and accuracy in handling the 
fundamental operations in arithmetic. 

Emphasis on the fundamental principles of arithmetic rather 
than on set rules for the solution of problems. 

Clearness and fullness of explanation. 

Problems that have an informational value. 

Attention is directed to the following features: 

The chapter on aliquot parts, as applied to billing, trade 
discount, and simple interest, is placed early in the text. 

Common and decimal fractions are treated interchangeably, as 
they are used in business. 

Problems are provided for mental work. 

Many problems are taken from actual business transactions of 
corporations, cities, states, and nations of the world. 

Calculation tables are illustrated and applied to the solution 
of problems. 

There are five 'sets of examinations (each set consisting of a 
speed test and a written test) to determine the student’s mastery 
of the subject at various stages of the work. 

Among the special features are the following: 

A rapid calculation drill chart in Aliquot Parts and Simple 
Interest. 

Suggestions and exercises for developing skill and accuracy in 
addition of simple numbers. 

A great variety of problems for practice work. 

The elimination of topics and problems which changing business 
and educational conditions warrant. 

3 


4 


PREFACE 


A full and complete treatment of such practical topics as pay 
rolls, graphs, trade discount, profit and loss, etc. 

Profits are ascertained when based on the sale price as well 
as on the cost price of goods. 

No attempt has been made to make New Complete Business 
Arithmetic a compendium of general business practice. On the 
other hand, an effort has been made so to present the subject of 
business arithmetic that those using the text shall be able to gain 
therefrom a knowledge of the principles governing business calcu¬ 
lations, together with a reasonable degree of skill in the use of such 
principles. 

The author wishes with much gratitude to acknowledge the 
kind criticisms of Complete Business Arithmetic given by teachers. 
He wishes also to express his appreciation of the assistance re¬ 
ceived from the many business men and firms who have so gener¬ 
ously given of their time and service in furnishing invaluable 
information about business practices and customs. 

Gborge H. Van Tuyl. 


CONTENTS 


PAGE 

Reading and Writing Numbers. 7 

United States Money .10 

Aliquot Parts.12 

Addition.38 

Subtraction.• • 50 

Multiplication.57 

Division.05 

Factoring.09 

Greatest Common D .visor.71 

Least Common Multiple.72 

Cancellation. 74 

Domestic Parcel Post.76 

Fractions.79 

Fundamental Principles of Arithmetic.116 

Average. 12 7 

Ratio and Proportion .128 

Examinations.134 

Denominate Numbers.138 

Metric System.143 

Involution and Evolution.102 

Mensuration.107 

Specific Gravity.191 

Practical Measurements .194 

Graphs . 22i 

997 

Examinations. 

229 

Percentage . 

Trade Discount. 250 

Profit and Loss. 2 ^3 


5 































6 


CONTENTS 


PAGE 

Manufacturing Costs.284 

Marking Goods.287 

Commission and Brokerage.292 

Examinations.299 

Interest.302 

Accurate.313 

Negotiable Paper.314 

Bank Discount.320 

Present Worth and True Discount.328 

Compound.332 

Sinking Funds.335 

Partial Payments..339 

Taxes.. 

Insurance.. 

Fire .353 

Life .360 

Savings Banks.304 

Postal Savings Banks.3 qq 

Examinations. 307 

Stocks. 37Q 

Bonds . 3g^ 

Exchange . 3gg 

Domestic. 3 qq 

Fore 'g n .. 

United States Customs. 4qq 

Equation of Accounts. 442 

Equation of Payments. 

Equation of Accounts. 4^5 

Cash Balance. 4^g 

Partnership. ^1 

Building and Loan Associations. 427 * 

Examinations. , 01 


































NEW COMPLETE 

BUSINESS ARITHMETIC 


READING AND WRITING NUMBERS 

1. Numbers are expressed in three ways: (1) by written woirds, 
(2) by letters (Roman system), and (3) by figures (Arabic system). 

ROMAN NOTATION 

2. In the Roman system of writing numbers seven letters are 
used. They are: 

I V X L C D M 

1 5 10 50 100 500 1000 

3. Writing numbers by the Roman system is based on the fol¬ 
lowing principles: 

Principles: 1. Repeating a letter repeats its value . 

2. The value of a letter written after a letter of greater value 
is to he added to the larger value. 

3. The value of a letter written before a letter of greater value 
is to he subtracted from the greater value. 

4. A horizontal bar over a letter increases its value one thou¬ 
sand times. 

Notes. (1) Only whole numbers can be written in the Roman system of 
notation. 

(2) Roman notation is of little practical value, being used only to record 
dates and chapters in books, dates on buildings, numbers on a watch face, etc. 

7 



8 


READING AND WRITING NUMBERS 


4. Roman Notation Illustrated 


II 

= 2 

XIV 

= 14 LIX 

= 59 

CCCLXVI = 366 

III 

= 3 

XVI 

= 16 LXXIV 

= 74 

DCXCVI = 696 

IV or IIII 

* = 4 

XIX 

= 19 LXXXIX = 89 

MCXIX = 1119 

VI 

= 6 

XXIII 

= 23 XC 

= 90 

MCM = 1900 

IX 

= 9 

XXIV 

= 24 CX 

= 110 

MCMX = 1,910,000 

5. l. 

Write in Roman notation: 




12 

85 

336 

598 

10,000 


18 

94 

472 

1077 

15,000 


21 

111 

505 

1384 

100,000 


47 

128 

666 

1527 

150,000 


63 

247 

872 

1898 

1,000,000 


2. Read the following: 


XVIII LXIX 

CDLIX 

MC 

XXIX XCIX 

DCD 

MDCCCXCIX 

XXXI CIX 

DCCCLXXXIX 

MDCCXXVIII 

XLIV CLXXII 

CM 

VCCLXXX 

XLIX CCCLX 

MCMIX 

L 


ABABIC NOTATION 

6. The Arabic system of notation makes use of ten characters 
or figures, and a period or decimal point. They are 0, 1, 2, 3, 4, 
5, 6, 7, 8, 9, and . 

7. With these figures and the decimal point any required num¬ 
ber can be written. 

8. For convenience in reading numbers, the figures are some¬ 
times divided into groups of three, in this way: 

2,564,879,598. 

The reason for this grouping is best understood by observing 
the following table: 


* IIII is used on timepieoes. 



ARABIC NOTATION 


9 


Table Showing how to Read Numbers 



The number is read, “eighty-nine trillion, seven hundred twenty- 
four billion, five hundred sixty-three million, nine hundred thou¬ 
sand, one hundred six.” 

9. In like manner read the following numbers: 


1. 21,460 

2. 34,829 

3. 127,724 

4. 306,200 

5. 425,002 

6. 800,005 


7. 962,024 

8. 872,463 

9. 900,047 

10. 1,456,789 

11. 3,248,764 

12. 5,002,005 


13. 27,250,820 

14. 20,300,600 

15. 65,897,249 

16. 185,496,728 

17. 593,972,847 


18. 15,647,989,721 


10. Write the following numbers, using the Arabic system: 

1. Seven hundred forty-three. 

2. Nine thousand fourteen. 

3. Eleven thousand one hundred four. 

4. Fifteen thousand twenty. 

5. Seventy-six thousand ninety-nine. 

6. Fourteen hundred eighty-eight. 

7. One hundred sixty thousand four hundred. 

8. Eight hundred thousand nine hundred ninety. 

9. Two million sixteen thousand seventeen. 

10. Six hundred eleven million fourteen thousand eighty-six. 


10 


READING AND WRITING NUMBERS 


UNITED STATES MONEY 


11. Any number of cents less than one hundred is read as 
cents: thus, $.45 or 45 i is read “forty-five cents,” but any value 
equal to or greater than one hundred cents is read as dollars, or as 
dollars and cents: thus, $8.75 or 875^ would be read “eight dol¬ 
lars, seventy-five cents.” A period, or point, separates the 
number of dollars from the number of cents. 


12. Read the following: 

$3.17 $12.40 $826.52 

$4.28 $16.50 $1624.43* 

$9.15 $85.90 $1837.16 

$10.20 $104.26 $2487.49 


$1476.83 

$1982.76 

$14,728.41 

$146,789.09 


13. Write the following in figures: 


l. One hundred sixteen dollars, fifty-nine cents. 


2. Three hundred nine dollars, seventy cents. 


3. Twelve hundred twenty-one dollars, eighty-eight cents. 

4. Twenty-four hundred forty-three dollars, nine cents. 

5. Fifty-seven hundred six dollars, thirty-three cents. 


14. Money is the generally accepted measure of value. 

15. Each nation has its own system of money or coinage. 


The principal nations of the world have adopted gold as the 
standard of monetary value. 

16. United States money consists of the coins, notes, and cer¬ 
tificates authorized by Congress to be used as money. It is a 
decimal system. 


Table 

10 mills = 1 cent, i 
10 cents = 1 dime, d. 
10 dimes = 1 dollar, $ 
10 dollars = 1 eagle 
(The mill is not a coin.) 


17. There are eleven different kinds of money in circulation in 
the United States: gold coins, silver dollars, subsidiary silver, 

* In reading $1624.43 say “sixteen hundred twenty-four dollars, forty-three cents,” not 
“one thousand six hundred twenty-four dollars, forty-three cents.” One reason for saying it 
and writing it as suggested is that it takes less time and space. 


UNITED STATES MONEY 


11 


nickels, pennies, gold certificates, silver certificates, Treasury 
notes, United States notes (greenbacks), Federal Reserve notes 
and Federal Reserve Bank Notes. 

18. The gold dollar, containing 25.8 grains of standard gold .9 
fine, is the unit of value. 

19. The silver dollar weighs 15.988 times as much as the gold 
dollar. That is, the ratio of gold to silver is 15.988 to 1 (com¬ 
monly called 16 to 1). 

20. Denominations, Weight, and Fineness of the Coins of 

the United States 


gold 


Denomination 

Fine Gold 
Contained 
(Grains) 

Allot 

Contained 

(Grains) 

Weight 

(Grains) 

One dollar not coined (since 1890) 

23.22 

2.58 

25.80 

Quarter eagle 

($ 2.50) 

58.05 

6.45 

64.5 

Half eagle 

($ 5.00) 

116.10 

12.90 

129. 

Eagle 

($10.00) 

232.20 

25.80 

258. 

Double eagle 

($20.00) 

464.40 

51.60 

516. 


SILVER 





Pure Silver 





Contained 



One dollar 


371.25 

41.25 

412.5 

Half dollar 


173.61 

19.29 

192.9 

Quarter dollar 


86.805 

9.645 

96.45 

Dime 


34.722 

3.858 

38.58 


MINOR 





Pure Copper 





Contained 



Five cents* * 

(nickel) 

57.87 

19.29 

77.16 

One centf 

(copper) 

45.60 

2.40 

48. 


Note. The alloy in gold and silver coins neither adds to nor detracts from 
the value of the coins. Its purpose is'to harden the coin, thus reducing the loss 
by abrasion. 

* 75% copper and 25% nickel. 


t 95% copper and 5% tin and zinc. 



























ALIQUOT PARTS 

21. An Aliquot part of a number is a number that is contained 
in it an integral number of times. 

Thus, 2, 4, 5, 10, 20, 25, and 50 are aliquot parts of 100; that is, 2 = °f 
100, 25 = j of 100, etc. 

Note. Other fractional parts of 100 are called aliquant parts of 100. For 
convenience, all fractional parts of 100 will be treated as aliquot parts. 

Halves and Quarters 

22. l. What part of 100 cents, or $1.00, is 25 50 75 

2. At 25 i a yard, how many yards can be bought for $1.00? 

3. If 4 yd. cost $1.00, how much will 8 yd. cost? 12 yd.? 
16 yd.? 

4. How much will 20 yd. cost at 25 i a yard? at 50 ^ a yard? 
at 75 i a yard? 

5. Compare 25 £ with 50 with 75 jzf. 50 £ with 75 jzf; with 

25 75 £ with 25 with 50 i. 

6. Compare the number of yards that can be bought for 
50 and for 25 ft; for 25 i and for 75 f; far 75 £ and for 
50 ff. 

7.. State a rule for finding the cost of any number of articles 
at 25 i each; at 50 each; at 75 ^ each. 

23. Make all extensions mentally, and find the total value of 
each of the following: 


1. 

2. 

3. 

16 yd. @ 25 i 

32 yd. @ 50 i 

120 yd. @ 25 

24 yd. @ 25 (4 

48 yd. @ 25 i 

140 yd. @ 50 

30 yd. @ 50 £ 

54 yd. @ 50 ^ 

150 yd. @ 25 

16 yd. @75(5 

60 yd. @ 75 

160 yd. @ 75 

40 yd. @50^ 

80 yd. @ 75 1 

126 yd. @ 25 


12 



ALIQUOT PARTS 


13 


4. 

13 yd. @ 25 i 
21 yd. @ 50 i 
39 yd. @ 25 i 
31 yd. @75 i 
48 yd. @75 i 


5. 

3486 lb. @ 25 i 
1984 lb. @ 50 i 
4821 lb. @ 2$ ff 
8448 lb. @ 75 ^ 
1331 lb. @ 50 i 


6 . 

128 lb. @ 75 i 
324 lb. @ 50 i 
233 lb. @ 25 i 
199 lb. @ 75 i 
247 lb. @ 25 i 


Make up and find the total value of each of two or more groups 
of quantities like the above as directed by the teacher. 


Eighths 


24. l. 12| i is what part of 25 2. \ of } is how much? 

3. 12! i is what, part of $1? 

4. 25 i is how many eighths of a dollar? 

5. 25 ^ +12= how many cents? 

6. 37 \ £ is what part of a dollar? 

7. What is the sum of \ and J? 50 + 12! i = ? 

8. 62! ([. is what part of a dollar? 9. 75 i + 12! = ? 

10. $f = how many cents? 

11. What part of a dollar is 12! j£? 37! ^? 62| ^? 87|^? 

12. What is the cost of 16 lb. of raisins at 12! i a pound? 
of 16 baskets of potatoes at 37! i a basket? of 16 yd. of 
cloth at 62| £ per yard? of 16 bu. of apples at 87 J ^ a 
bushel? 


25. Using the aliquot-part method, find the total value of each 
of the following: 


1 . 

24 lb. @ 37! i 
32 lb. @ 12| i 
40 lb. @ 62J $ 
56 lb. @ 37J i 


2 . 

48 lb. @ 12! i 
64 lb. @ 25 i 
72 lb. @ 37! i 
88 lb. @ 87! i 


3. 

120 yd. @ 12! ^ 
164 yd. @ 12! i 
200 yd. @ 37! i 
240 yd. @ 12! i 


Make two or more groups like the above and find the total value 
of each. 


14 


ALIQUOT PARTS 


Sixteenths 

26. 1 . 6J £ is what part of 12§ ff? of 25 jf? of 50 jzf? of $1? 

2. 3 times 6J £ is how many cents? 

3. 18f i is what part of a dollar? 

4. 25 i or is how many sixteenths of a dollar? 

5. 25 i + 6| £ are how many cents? 

6. 31J 0 is what part of a dollar? 

7. | — ■£$ = ? 43f i is what part of a dollar? 

8. \ +i 3 6 = ? 56J i = what part of a dollar? 

9. In like manner find how many cents there are in Sri; 
$y§- 7 and $xf by comparing 6J t (Srg-) with lb £ and SI. 

27. Study the following table and compare values with one 
another. 


Halves 

Quarters 

Eighths 

Sixteenths 

1 . 



.06* 

2. 


• 12| 


3. 



.18f 

4. 

.25 



5. 



.311 

6. 


Hoi 

CO 


7. 



.431 

8. .50 




9. 



.561 

10. 


.62* 


11. 



.681 

12. 

.75 



13. 



.811 

14. 


.87* 


15. 



.93f 

16. 1.00 

1.00 

1.00 

1.00 


28. In the above table, note that the number of sixteenths is 
indicated in the left hand column. Thus, .43J is on the 7th line, 
hence .43f is .62J is on the 10th line, hence it is or f, etc. 












ALIQUOT PARTS 


15 


Note that .31}, being on the 5th line, is is the sum of T \ 

(1) + ts (} of tV). That is, to find the value of 492 yd. at 31} ^ 
solve thus: 


$492 


value at 


123 

30 


=value at .25 

75 = value at .06} 


$153 75 = value at $ .31J 


Find the value at $.25 (ye) 
and then at $.06} and add 
the results. 


If, in the above illustration, the price had been $-18}, the 
operation would have been the same, except the value at $.06} 
would be subtracted from the value at $.25. 

Again, from the table, note that .81} is on the 13th line, hence 
.81} equals yf. yf =t 8 6 +tV +y§-, or -81} =-50 +.25 +.06}. 

Hence, to find the value of 578 yd. at .81} t, solve thus: 


$578 

=value at $1. 

289 

=value at 

.50 

144 

50 = value at 

.25 

36 

13 = value at 

.06} 

$469 

63 = value at $ 

bo 

i— 1 

iHh* 


To multiply by .93} (}£), simply deduct y^. 

Similarly, analyze .56}; .43}; .68}, and apply the above method 
of solution to the following exercises, unless results can be deter¬ 
mined mentally. 

29. Find the total value of each of the following: 


1 . 


32 yd. @ 6} i 

80 yd. 

48 yd. @ 12} i 

128 yd. 

64 yd. @ 18} i 

144 yd. 

96 yd. @ 6} i 

96 yd. 

160 yd. @ 43} i 

72 yd. 

80 yd. @ 31} i 

112 yd. 

136 yd. @ 56} £ 

152 yd. 

72 yd. @ 18} £ 

84 yd. 


2. 

3. 

@311*5 

47 yd. @25 i 

@37i *5 

53 yd. @ 12} i 

@;56 |*5 

58 yd. @ 6} i 

@93f,5 

63 yd. @ 50 ^ 

@371*5 

75 yd. @ 37} i 

@371*5 

104 yd. @ 18} i 

@311*5 

168 yd. @ 87} i 

@ 181*5 

192 yd. @ 75 £ 








16 


ALIQUOT PARTS 


4. 

194 yd. @ 87} £ 
64 yd. @ 18f ff 
552 yd. @ 81} j* 
104 yd. @ 62} ^ 
4264 yd. @ 6} i 


5. 

39 yd. @ 56} ff 
96 yd. © 31} f* 
1046 yd. @ 87} £ 
52 yd. @ 43} £ 
1972 yd. @ 93} j* 

Thirds and Sixths 


6 . 

108 yd. @ 68} £ 
3922 yd. @ 37} i 
3360 yd. @ 18} ^ 
5497 yd. © 81} ^ 
2849 yd. © 56} p 


30. l. What part of a dollar is 33} ff? 66f ? 

2. What is } of }? } of 33} i is how many cents? 

3. 16§ £ = what part of a dollar? 

4. 3 times 16 fjzf = how many cents? what part of a 
dollar ? 

5. 4 times 16§ ^ = how many cents? what part of a 
dollar? 

6. $1 — 16§ f, = how many cents? what part of a dollar? 


31. Find total value of each group: 


1 . 2 . 

48 lb. @ 16f 66 yd. @ 

54 lb. @ 66} i 76 yd. © 

30 lb. @ 83} i 76 yd. © 

4. 5. 

64 lb. © 6} i 48 qt. © 

96 lb. @ 33} i 72 qt. @ 

88 lb. @ 37} i 96 qt, @ 

91 lb. @ 66| i 60 qt. @ 

Write and find the total value 
above. 


3. 


mi 

57 yd. @ 66§ £ 

25 i 

72 yd. @ 16} i 

mi 

90 yd. @ 83} £ 

33§ i 

6. 

120 yd. © 37i i 

i6| i 

126 yd. @ 16| i 

mi 

132 yd. @ 83| i 

6i i 

144 yd. © 33§ i 

of two or 

more groups like the 


Twelfths 


32. l. } of 25 ^ = how many cents? 

2. } of } = ? 8} £ = what part of a dollar? 

3. 50 £ — 8} £ = how many cents? 

4. } — T* = ? $41} i = what part of a dollar? 

5. 50 i + 8} i = how many cents? what part of a dollar? 

6. $1 — 8} £ = how many cents? what part of a dollar? 


ALIQUOT PARTS 


17 


33. Study the following table and compare values with one 
another. 


Halves 

Thibds 

Quarters 

Sixths 

Twelfths 

1. 




.08! 

2. 



.16! 


3. 


.25 



4. 

.331 




5. 




.41! 

6. .50 





7. 




.58! 

8. 

•66f 




9. 


.75 



10. 



.83! 


11. 




.91! 

o 

o 

cl 

«— i 

1.00 

1.00 

1.00 

1.00 


34. Analyze .4If. 

.41§, being on the 5th line, is i^-. T 5 2 is ^ less than (i). 

That is .41§ equals .50 less \ of .50. 

Similarly, analyze .58J. 

35. Using the above suggested method of solution where 
advantageous, find the total value of the following groups: 


1 . 

36 yd. @ 8} 

48 yd. @ 16§ i 
24 yd. @ 41§ t 
60 yd. @ 58J i 
72 yd. @ 83J i 

Note. 
knowledge 


2 . 

66 yd. @ 33^ i 
84 yd. @ 41| ^ 
120 yd. @91U 
42 yd. @ 58i i 
54 yd. @ 16f i 


3. 

96 yd. @ 83i i 
84 yd. @ 41f ff 
72 yd. @ 91f ^ 
60 yd. @ 


132 yd. @ 58i i 

Many articles are bought by the dozen and sold by the piece. A 
of the “twelfths” is essential in determining the price of one. 


4. 

75 lb. 

88 lb. @ 16§ i 
90 lb. @ 41| i 
961b. @ 58 it 
84 lb. @ 66f £ 


5. 

24 yd. @ 58} £ 
36 yd. @ 41f ff 
60 yd. @ 91f ^ 
48 yd. @ 16f £ 
72 yd. @ 33} ^ 


6 . 

140 lb. @ 

160 lb. @ 91f 
240 lb. @ 58} 
360 lb. @ 41} l 
118 lb. @/ 8} ^ 


VAN TUYL’S NEW COMP. AR.—2 














18 


ALIQUOT PARTS 


REVIEW 

36. Using the shortest solution, find the total cost of: 


1 . 

54 yd. @ 16f £ 
40 yd. @25*4 
36 yd. © 33} £ 
84 yd. © 50 £ 

75 yd. © 66§ £ 
96 yd. © 121 £ 
48 yd. © 8} (4 
36 yd. © 16| £ 
42 yd. @ 25 £ 

4 . 

76 yd. © 37} £ 
92 yd. @ 31} £ 
83 yd. © 43} £ 
78 yd. @ 41f £ 
107 yd. © 6} (4 

113 yd. © 62§ 0 

114 yd. @ 83} £ 
131 yd. @ 16| £ 
125 yd. @ 25 £ 
143 yd. @ 6} £ 

7 . 

570 lb. © 93} £ 
629 lb. @ 91f £ 
328 lb. © 87} i* 
245 lb. @8U 
446 lb. @ 68} £ 
928 lb. @ 75 £ 
476 lb. @ 18} £ 
846 lb. © 56} (4 
382 lb. @ 58} £ 
593 lb. @ 311 


2 . 

120 yd. @ 62} £ 
240 yd. @ 87} £ 
160 yd. @ 66§ £ 
360 yd. @ 91§ £ 
164 yd. © 371 (4 
66 yd. @ 41f £ 
68 yd. © 43} £ 
75 yd. @ 58} £ 
111yd. @33i £ 

5 . 

78 lb. @ 18} £ 
93 lb. @ 25 £ 

82 lb. @ 33; (4 
13 lb. @ 31} £ 

9 lb. © 37§ £ 
27 lb. @ 41f £ 
32 lb. © 43} £ 
84 lb. @ 50 £ 

79 lb. @ 56-1 y! 
97 lb. @ 58} £ 

8 . 

234 lb. @ 87} (4 
859 lb. @ 43} £ 
720 lb. @ 81i ji 
924 lb. @ 33} £ 
634 lb. © 93} £ 
210 lb. @ 58} )4 
446 lb. @ 12} ^ 
888 lb. @ 62} 
356 lb. @ 56} £ 
926 lb. @ 31} £ 


3 . 

33 yd. © 12} (4 
41 yd. @S\£ 
52 yd. @ 6} £ 
63 yd. © 12} £ 
89 yd. © 33} j4 
113 yd. @75ji 
97 yd. © 62} 
116 yd. @87} (4 
121 yd. @ 56} £ 

6 . 

87 lb. @ 6} £ 
94 lb. @ 8} £ 

104 lb. @ 12} £ 
231 lb. @ 16| £ 
321 lb. © 18} £ 

88 lb. @ 93} £ 
1361b. @91f(4 
316 lb. © 83} ^ 

28 lb. @ 81} £ 
7 lb. @ 68} £ 

9 . 

448 lb. @ 56} £ 
234 lb. @ 6} £ 
756 lb. @ 81} £ 
335 lb. © 91§ £ 
543 lb. @ 58} £ 
636 lb. © 62} £ 
927 lb. @ 66| £ 
360 lb. © 83} £ 
448 lb. @ 91f (4 
864 lb. © 68} £ 


ALIQUOT PARTS 


19 


Sevenths, Ninths, and Fifteenths 

37. In addition to the aliquot parts of a dollar already found, 
there are a few others of which use is occasionally made: viz. 
sevenths, ninths, and fifteenths. 


1. What is y of a dollar? f? f? 

2. What is i of a dollar? f? f ? 

3. What is Yt of a dollar? ^? 

Note. These lists will not be made complete, as there is seldom any use 
for others than those given above. 


38. Make the extensions mentally, and find total value of each 


group: 

l. 


21 lb. @ 14f i 

42 lb. 

28 lb. @ 28y i 

54 lb. 

36 lb. @ 11 it 

60 lb. 

45 lb. @ 6§ i. 

84 lb. 

48 lb. @ 6^ 

91 lb. 


2. 

3. 

@42H 

33 lb. @ 14f i 


451b. @281?! 

© 13§ i 

83 lb. @ 6| i 

@ 

961b. @ 111?! 

©mi 

87 lb. @ 121 ?! 


Write two or more groups and find total value of each. 

39. In the preceding exercises the price has been an aliquot 
part of $1. The same principle applies when the quantity is an 
aliquot part of 100 units of the article involved. 

The cost of 48 yd. @ 12J $ = the cost of 12J yd. @ 48 ff. 

(a) 48 yd. @ 12! = ! of 48 = 6, hence $6, cost. 

(b) 100 yd. @ 48 i = $48. 

12| yd. = | of 100 yd. 

Hence, 12! yd. @ 48 £ cost ! of $48 = $6, cost. 

40. Find total value of each of the following groups: 


l. 

25 yd. @ 32 i 
331 yd. @ 45 i 
50 yd. @ 78 ?! 
37iyd. @48(4 
75 yd. @ 80 i 


2 . 

16| yd. @ 42 i 
18f yd. @ 56 jS 
311 yd. @80^ 
62J yd. @ 72 i 
93f yd. @ 96 i 


3. 

6} lb. @ 16 i 
811b. @ 24 
121 lb. @ 32 i 
16§ lb. @ 42 i 
18f lb. @ 48 1 


20 


ALIQUOT PARTS 


4. 

25 lb. @ 43 i 
31i lb. @ 56 i 
33} lb. @ 58 i 
37} lb. @ 68 i 
43} lb. @ 96 6 

7. 

89 lb. ©m 
93 lb. @ 6} i 
87 lb. @ 12J ^ 
76 lb. @ 16§ 
116 lb. @25^ 


5. 

78 yd. @ 33J i 
16| yd. @ 33 i 
56} yd. @ 44 l 
581 yd. @ 66 £ 
38 yd. @ 621 £ 

8 . 

160 yd. @ 75 ^ 
200 yd. @ 871 £ 
240 yd. @ 93} i 
25 yd. @ $1.20 
50 yd. @ $1.60 


6 . 

43 yd. @ 66f £ 
68} yd. @ 32 ^ 
81} yd. @16^ 
83} yd. @ 72 
80 yd. @ 93} ^ 

9. 

1148 yd. @33}jzf 
2682 yd. @ 371 
81} yd. @ 80 £ 
68} yd. @ 96 ^ 
75 yd. @ 72 ^ 


APPLICATION OF THE PRINCIPLES OF ALIQUOT PARTS 
41. l. Multiply 48 by 10; by 100; by 1000. 

2. What did you do to multiply by 10? by 100? by 1000? 

3. 2} is what part of 10? 25 is what part of 100? 250 is 
what part of 1000? 

4. Multiply 48 by 2}; by 25; by 250. 

First multiply 48 by 10: that is, annex one cipher, thus, 480. Since the 
given multiplier is 1 of 10, the required product is \ of 480, or 120. 

In like manner to multiply by 25, first annex two ciphers to 48, thus, 4800, 
and divide by 4, because 25 is 1 of 100, etc. 

5. 3} is what part of 10? 

33} is what part of 100? 

333} is what part of 1000? 

6. Multiply 144 by 3}. 

Since 31 is 1 of 10, first annex one cipher (multiply by 10), thus, 1440, and 
divide by 3. The required product is 480. 

7. 8. 

1} is what part of 10? If is what part of 10? 

12} is what part of 100? 16f is what part of 100? 

125 is what part of 1000? 166§ is what part of 1000? 


ALIQUOT PARTS 


21 


42. Copy and complete the following table: 



l 

10 

100 

1000 

I’s 

.12| 

It 

12* 

125 

i’s 





Vs 





Vs 






43. Make the extensions mentally and find the total value of 
the items in each group: 


1 . 2 . 


24 

A. 

@ 

$25 

25 

A. © 

$96 

18 

cows 

@ 

$25 

33* 

bu. @ 

$1.08 

72 

sheep 

@ 

$2* 

166* 

bu. @ 

$.84 

16 

plows 

@ 

$12* 

1250 

bu. @ 

$.64 

32 

horses 

@ 

$125 

2500 

bu. @ 

$.78 

160 

bu. 

@ 

$1* 

1600 

bu. @ 

$1.25 

330 

bu. 

@ 

$1.66| 

16} 

bu. @ 

$1.44 

48 

wagons 

@ 

$125 

3800 

lb. @ 

$.12* 


3. 4. 


125 bu. 

© $1.50 

43J 

bu. 

@ $.64 

75 wagons @ $83* 

520 

lb. 

© $.18} 

6* lb. 

© $.44 

66 

lb. 

@ $.58* 

2500 lb. 

© $.39 

750 

plows 

© $16 

12* bu. 

@ $.98 

87* 

bu. 

© $1.84 

35 horses 

@ $250 

1500 

bu. 

© $2.60 

175 sheep 

@ $12.50 

78 

wagons 

© $166* 

50 cows 

@ $137.50 

438 

lb. 

@ $.06* 

5. 

8* bu. 

@ $1.50 

225 

6. 

bu. 

@ $2.50 

29 cows 

@ $125 

40 

horses 

@ $250 

59 sheep 

@ $16} 

124 

desks 

@ $50 

500 lb. 

@ $.19 

43} 

lb. 

@ $.36 

35 desks 

© $75 

1500 

lb. 

@ $.44 

66§ bu. 

© $.90 

17 

cows 

© $125 

580 lb. 

© $.18f 

464 

sheep 

@$18} 

25 lb. 

@ $9 

28 

chairs 

@ $33* 














22 


ALIQUOT PARTS 


TRADE DISCOUNT 

44 . All the information about aliquot parts in the preceding 
pages is directly applicable to Trade Discount. 

45 . The expression “per cent” (symbol %) means hundredths. 
That is, 25% = Tini = b 

46 . Find the net cost of the following, allowing a discount of 
374%: 

66 lb. @ 66f i = $44 
80 lb. @ 40 1 = $32 

120 lb. @ 75 i = $90 
240 lb. @ 374 i = S90 $256 

Less 374% (1 + 4 of 4) _96 $160 

The items are extended mentally by taking f of 66; f of 80; f of 120; and 
| of 240. By adding, the gross cost is found to be $256. 37^% discount = a 

discount of f (1 + £). f of $256 = $96, which deducted, leaves $160, the net 
cost. 


47 . Find the net cost of each of the following: 


l. 

72 lb. @ 16f t 
48 lb. @ 64 t 
32 lb. @ 124 i 
56 lb. @ 374 i 
Less 124% 

4. 

48 doz. @ 64 i 
64 doz. @ $1.25 
24 doz. @ 84 ff 
374 doz. @ $2.40 
Less 334% 

7. 

196 doz. @ $1.25 
26 doz. @ 75£ 

86 doz. @ 124 i 
50 doz. @74 f 
Less 25% 


2. 

72 lb. @ 25 ^ 

96 lb. @ 334 i 
90 lb. @44f i 
80 lb. @ 874 1 
Less 18f% 

5. 

64 lb. @ 124 i 
240 lb. @ 334 i 
480 lb. @ 564 i 
320 lb. @ 64 i 
Less 10% 

8 . 

326 doz. @ 624 1 
112 doz. @75^ 
84 doz. @ 624 i 
2500 lb. @ 64 i 
Less 374% 


3. 

36 doz. @ 75 i 
48 doz. @ 124 i 
60 doz. @25 i 
72 doz. @ 374 t 
Less 16f% 

6 . 

125 lb. @ 48 i 
375 lb. @ 72 i 
320 lb. @ 624 1 
250 lb. @ 36 i 
Less 124% 

9 . 

841 lb. @ 124 i 
963 lb. @ 16J i 
1047 lb. @ 25 i 
141 lb. @ 334 l 

Less 20% 


SHORTENING MULTIPLICATION 


23 


FURTHER SUGGESTIONS FOR SHORTENING MULTIPLICATION 

48 . In addition to the so-called Aliquot Parts of 100, the use 
and application of which have been explained, there are many 
other numbers that are a fractional part more or less than the 
aliquot parts. By making use of this fact the work of multiplying 
by numbers that are not aliquot parts of 100 may be shortened. 
For example, 13} ft is not an aliquot part of a dollar, yet to mul¬ 
tiply by .13} requires no actual multiplication. 

1. Multiply 120 by .13}. 

.134 = .10 + (} of .10) 

Therefore, to multiply 120 by .13|, point off one place in 120, thus, 12.0, 
and add f of 12, which gives a result of 16. 

2. Multiply 36 by .23}. 

♦23} = 20 + (} of .20) - * + (* of A) 

3.6 = T V of 36 . 

7.2 = A of 36. 

L2 = } of A of 36 
8.4 = .23} of 36 

Moving the point one place to the left in 36 gives ttt of 36; multiplying by 
2 gives t 2 (7 of 36. To find the remaining part take £ of 7.2, which is 1.2. Add 
this to 7.2, giving 8.4, the desired result. 

3. To multiply by .13f, find value at .12} and add A of 
the result. 

That is, to find the value of 104 yd. at 13J ft, solve thus: 

$104| = value at SI 


131 = value at 

1|30 = value at 


. 12 } 

.01} (tV of $.12}) 


14|30 = value at S .13f 

To multiply by .27}, find the value at .25 and add T V of 
the result. 

That is, to find the value of 146 yd. at 27} ft, solve thus: 


f S146 

= value at $1 

36 

50 = value at .25 

3 

65 = value at .02} 

S 40 

15 = value at $ .27} 








24 


ALIQUOT PARTS 


5. To multiply by .33 J, point off 1 place, multiply by 3, and 
add J of the result. (A vertical line takes the place of 
the decimal points.) 

Thus 426 X $.33J: 


$ 42 

6 

= value at $.10 

127 

8 

= value at 

.30 

15 

975 

= value at 

•03| 

143 

775 



143 

78 

= value at $.33| 


6 . To multiply by .35, find the value at .25 and at .10, and 
add the results. 

7. To multiply by .45, find the value at .50, and deduct 
■nr of the result. 

8 . To multiply by .83^; . 874 ; *91§ ; .93f or other fractional 
values which lack but one part of unity, deduct the one 
part from the value at unity. 

Thus, 137 X S.83J. 

$137| = value at $1. 

22|83 = value at .16§ 

114| 17 = value at .83§ 

Note. This plan of solution avoids the necessity of retaining the fraction 
in the value at $.16f. .83 £ is 5 times .16f. If $22.83 is multiplied by 5, the 

result is $114.15, which is incorrect, due to the dropping of the fraction £ in 
the value, $22.83. 

SUGGESTIONS FOR CLOSE OBSERVATION AND PRACTICE 

9. To multiply by .02§, point off 1 place and divide by 4- 

10. To multiply by .03§, point off 1 place and divide by 3. 

11. To multiply by .05, point off 1 place and divide by 2. 

12. To multiply by .07|, point off 1 place and deduct J of 
the result. 

13. To multiply by .09, point off 1 place and deduct T V of 
the result. 

14. To multiply by .10, point off 1 place. 

15. To multiply by .11, point off 1 place and add ^ of the 
result. 






SHORTENING MULTIPLICATION 


25 


16. To multiply by .Ilf, point off 1 place and add f of the 
result. 

17. To multiply by .Ilf, divide by 8, and deduct .01 of the 
original number. 

18. To multiply by .13J, divide by 8 and add .01 of the 
original number. 

19. To multiply by .15, point off 1 place and add f of the 
result. 

20. To multiply by .17f, point off 1 place, add § of the result 
and \ of the second result. (.17f = .10 + .05 + .02f) 

21. To multiply by .18, point off 1 place, multiply by 2 and 
deduct iV of the result. 

22. To multiply by .19, point off 1 place, multiply by 2 and 
deduct .01 of original number. 

23. To multiply by. 22f, divide by 4 and deduct T V of the 
result. 

24. To multiply by .23f, divide by 4 and deduct of the 
result. 

25. To multiply by .24, divide by 4 and deduct .01 of the 
original number. 

26. To multiply by .26, divide by 4 and add .01 of the 
original number. 

27. To multiply by .26f, point off 1 place and add -g- of the 
original number. (.26f = .10 + .16§) 

28. To multiply by .35, point off 1 place and add f of the 
original number, etc. 

These few suggestions are made here in the hope that the 
student, after becoming familiar with them, will be led to discover 
many other new and related combinations for himself. 

49. Find total value of each group: 


l. 


48 yd. @ 7$ i 

90 yd. 

56 yd. @ 9 * 

96 yd. 

64 yd. @ 13J * 

110 yd. 

80 yd. @ 11$ * 

116 yd. 

88 yd. @ 17$ * 

124 yd. 


2. 

3. 

@18** 

7£yd. @36f! 

@22 <t 

11$ yd. @48* 

@22 \<t 

16 yd. @ 45 * 

@27U 

17$ yd. @62* 

@35** 

18 yd. @ 65 * 


26 


ALIQUOT PARTS 


4. 

5. 

6. 

55 yd. @ 44 i 

55 yd. @ 36 i 

65 yd. @ 33 i 

51 yd. @ 27 |4 

39 yd. @ 55 i 

132 yd. © 22 ^ 

141 yd. @ i 

123 yd. @ 17* f4 

163 yd. @ 27* f! 

172 yd. @ 27 i 

2191 yd. @ 33 0 
3131 yd. @ 35 ^ 
3171 yd. @44^ 
2731 yd. @ 45 
124} yd. @ 55 ^ 

7. 

8. 

9. 

24 lb. @ 2i i 

24 lb. @ 3| i 

48 lb. @ 5 i 

44 lb. @ 74 i 

56 lb. @ 9 i 

123 lb. @ 10 i 

113 lb. @11)4 

120 lb. @ ll|fS 

140 lb. @ 12 i 

150 lb. @ 13* )4 

84 lb. @ 15 i 

92 lb. @ 171 i 
64 lb. @ 22 i 

72 lb. @ 23f i 
44 lb. @ 24 ^ 

10. 

ll. 

12. 

54 lb. @ 26§ i 

88 lb. @ 271 i 

144 lb. @ 31J fS 

160 lb. @ 33f t 

176 lb. @ 35 i 

37 by 11 

28 by 13J 

44 by 18J 

48 by 22 

56 by 22* 

140 by 311 

150 by 371 

160 by 43f 

180 by 45 

190 by 471 

13. 

14. 

15. 

465 yd. @ 18 i 

532 yd. @ 27§ i 
651 yd. @ 33f 

576 yd. @ 45 i 

733 yd. @ 36 jl 

563 yd. @ lljji 
871 yd. @ 13f i 
728 yd. @ 17* (4 
193 yd. @22 (4 

292 yd. @ 22* (4 

364 lb. @ 2! i 
573 lb. @ 31 i 
492 lb. @ 5 i 
532 lb. @ 71 i 
657 lb. @ 9 i 

16. 

17. 

18. 


654 bu. @ $2.50 
832 bu. @ $1.25 
960 bu. @ $.875 
884 bu. @ $1.12| 
762 bu. @ $3.37| 


847 bu. @ $1.15 
933 bu. @ $1.35 
538 bu. @ $2.25 
854 bu. @ $1.55 
732 bu. @ $3.25 


496 A. @ $125 
534 A. @ $112.50 
678 A. @ $87.50 
742 A. @ $250 
232 A. @ $137.50 


RAPID DRILL EXERCISE 


27 


RAPID CALCULATION DRILL CHART 

50 . Suggestion to Teachers. Let the pupils choose sides as in 
a spelling match. Read the following exercises to them and have 
each student in turn state the results. Keep a record of the credits 
for each side, or let pupils choose from opposite side when a 
credit is made. Other ways of using the chart will be devised by 
the teacher to suit the particular needs of the class. 


16 

yd. 

@ 

25 

i 

43 

yd. 

@ 

25 i 

37§ 

yd. 

@ 

48 0 

32 

yd. 

® 

25 

i 

49 

yd. 

@ 

50 £ 

62} 

yd. 

@ 

32 i 

24 

yd. 

@ 

25 

i 

56 

yd. 

@ 

75 (4 

12} 

yd. 

@ 

56 £ 

32 

yd. 

@ 

50 

i 

38 

yd. 

@ 

25(4 

m 

yd. 

@ 

72 £ 

44 

yd. 

@ 

25 

i 

41 

yd. 

@ 

25 i 

37} 

yd. 

@ 

64 i 

52 

yd. 

@ 

25 

i 

65 

yd. 

® 

50 (4 

32 

yd. 

@ 

6*?! 

48 

yd. 

® 

50 

i 

99 

yd. 

® 

25(4 

48 

yd. 

@ 

6* ?! 

36 

yd. 

@ 

25 

i 

97 

yd. 

@ 

50 (4 

64 

yd. 

@ 

181(4 

44 

yd. 

@ 

75 

i 

91 

yd. 

@ 

25(4 

64 

yd. 

@ 

31 \i 

24 

yd. 

@ 

75 

i 

90 

yd. 

® 

50^4 

32 

yd. 

® 

31*?! 

36 

yd. 

® 

50 

i 

89 

yd. 

® 

25 (4 

24 

yd. 

@ 

6* (4 

48 

yd. 

® 

75 

i 

32 

yd. 

@ 

12$ (4 

48 

yd. 

@ 

31*?! 

60 

yd. 

@ 

75 

i 

48 

yd. 

@ 

12* (4 

32 

yd. 

@ 

18f(4 

64 

yd. 

@ 

25 

i 

32 

yd. 

@ 

37* (4 

16 

yd. 

@ 

6* ?! 

68 

yd. 

® 

25 

i 

24 

yd. 

@ 

37* (4 

32 

yd. 

@ 

56* (4 

64 

yd. 

® 

50 ^ 

16 

yd. 

@ 

62* (4 

48 

yd- 

@ 

56* i 

84 

yd. 

@ 

25 

i 

24 

yd. 

@ 

62* (4 

6i 

yd. 

® 

32?! 

88 

yd. 

@ 

50 

i 

32 

yd. 

@ 

87* fS 

18J 

yd. 

@ 

32(4 

88 

yd- 

® 

75 

i 

48 

yd. 

@ 

37* (4 

6J 

yd. 

@ 

48(4 

96 

yd. 

® 

75 

i 

52 

yd. 

@ 

12* (4 

6i 

yd. 

® 

64(4 

92 

yd- 

@ 

50 

i 

64 

yd. 

@ 

37*?; 

18} 

yd. 

@ 

32?! 

25 

yd- 

@ 

16 

i 

64 

yd. 

@ 

62* (4 

18 

yd. 

@ 

33* (4 

50 

yd. 

® 

24 

i 

72 

yd. 

@ 

12*?4 

24 

yd. 

® 

33* (4 

25 

yd- 

® 

36 

i 

80 

yd. 

@ 

37* (4 

27 

yd. 

® 

66|?! 

75 

yd. 

® 

44 

i 

88 

yd. 

@ 

62*?; 

33 

yd. 

@ 

66|?4 

25 

yd. 

® 

56 

t 

36 

yd. 

@ 

12*?; 

36 

yd. 

® 

33* (4 

50 

yd. 

@ 

63 

l 

44 

yd. 

® 

12* (4 

42 

yd. 

@ 

33* (4 

25 

yd. 

® 

22 

i 

48 

yd. 

® 

62*?! 

48 

yd. 

@ 

66* (4 

25 

yd. 

@ 

54 

i 

56 

yd. 

@ 

37*?! 

21 

yd. 

@ 

66* (4 


28 


ALIQUOT PARTS 

RAPID CALCULATION DRILL CHART 


12 yd. @ 831 i 
78 yd. @ 16| i 
72 yd. @83H 
43 yd. @ 16f ^ 
54 yd. @ 83J t 
32 yd. @ 16f ^ 
27 yd. @ 16| 
60 yd. @ 831 
84 yd. @ 16| £ 
24 yd. @ 831 * 
30 yd. @ 831 ^ 
42 yd. @ 16f £ 
36 yd. @ 831 ^ 
42 yd. @ 831 * 
48 yd. @ 331 * 
48 yd. @ 16| £ 
45 yd. @ 16f* 
51 yd. @ 16f ^ 
54 yd. @ 16f ^ 
60 yd. @ 831 ^ 
66 yd. @ 831 £ 
72 yd. @ 16H 
24 yd. @ 81 i 
36 yd. @ 831 £ 
48 yd. @ 41f t 
24 yd. @ 41| i 
36 yd. @ 41f ^ 
48 yd. @81^ 
60 yd. @ 41f £ 
60 yd. @ 581 ^ 
72 yd. @81^ 
84 yd. @ 81 ff 
24 yd. @ 581 i 
36 yd. @ 581 1 
12 yd. @ 81 i 
12 yd. @ 41f £ 


24 A. @ $331 
32 A. @$25 

63 A. @ $331 
40 A. @ $25 
32 A. @ $125 
24 A. @ $25 
16 A. @ $250 
24 bu. @ $1,331 
36 bu. @ $2.50 
44 bu. @ $1.25 
52 bu. @ $2.50 
48 bu. @ $1.25 
56 T. @ $12.50 

64 T. @ $12.50 
72 T. @ $12.50 
16 oz. @ 21 ff 
24 oz. @ 21 ^ 

21 oz. @ 31 ^ 

28 oz. @ 21 

32 oz. @ 21 £ 

36 oz. @ If ^ 

33 oz. @ 31 ^ 

48 oz. @ 11 ^ 

56 oz. @21 £ 

12 oz. @ 31 
16 oz. @ 11 ff 
32 oz. @ 11 ^ 

32 oz. @ 21 i, 

48 oz. @ If i 
48 oz. @ 31 ^ 

66 oz. @ 31 ^ 

48 oz. @ 25 i 
56 yd. @ 121 £ 
64 yd. @ 371 i 
72 yd. @ 621 i 
84 yd. @ 25 £ 


72 1b. @ 331 i 
66 1b. @66}* 
54 1b. @ 66f ff 
72 1b. @81^ 

60 1b. @75^ 

66 1b. @831^ 
84 lb. @ 16f £ 
90 lb. @ 16J £ 
96 1b. @ 66f i 
48 yd. @ 581 i 
60 yd. @ 41§ ± 
32 yd. @ 621 i 
48 yd. @ 561 i 

39 yd. @ 331 1 
72 yd. @ 831 i 
27 oz. @ 31 i 
26 oz. @ 21 i 
96 oz. @ 11 £ 

32 oz. @ 21 i 
96 oz. @21 jt 
96 oz. @ 310 
96 bu. @ $1.25 
24 bu. @ $1.66f 
80 T. @$61 
72 T. @$81 
42 T. @ $121 
30 T. @$12.50 
42 T. @ $16§ 
72 bu. @ $1.66} 
24 bu. @ $2.50 
72 yd. @ 331 i 
64 yd. @ 75 i 
32 yd. @ 561 i 

40 yd. @ 621 i 
36 yd. @ 41§ i 
20 lb. @ 121 i 


ALIQUOT PARTS 


29 


ALIQUOT PARTS APPLIED TO DIVISION 

51 . l . At 25 £ a yard, how many yards of cloth can be bought 
for $1? for $2? for $10? for $25? 

2. At 37§ £ a yard, how many yards can be bought for $1? 
for $2? for $5? for $15? 

37! = $f. $1 -5- $! = f; f yd., or 2f yd., can be bought for $1. 

Since f yd., or 2f yd., can be bought for $1, 2 times 2f yd., or 51 yd., can be 
bought for $2, and 5 times 2| yd., or 131 yd., can be bought for $5, etc. 

Note. 1 -r | = f. f is called the reciprocal of |. The reciprocal of any 
fraction is the fraction inverted. 

Hence, to divide by an aliquot part, multiply by the reciprocal 
of the fractional equivalent. 

52. Find the number of yards that can be bought in each case: 


Total Cost 

Cost op One Yard 

Total Cost 

Cost of One Yard 

1 . 

$15.00 

$ .33! 

16. 

$ 91.00 

$ .58! 

2. 

36.00 

.75 

17. 

100.00 

.62! 

3. 

75.00 

.62| 

18. 

110.00 

•66f 

4. 

25.00 

.31| 

19. 

121.00 

•68f 

5. 

13.00 

‘.14* 

20. 

129.00 

.75 

6. 

16.00 

08| 

21. 

130.00 

.81! 

7. 

27.00 

.18* 

22. 

135.00 

.83! 

8. 

29.00 

.25 

23. 

59.50 

.87! 

9. 

35.00 

.31! 

24. 

66.00 

.91! 

10. 

41.00 

.33! 

25. 

75.00 

.93! 

11. 

42.00 

.37! 

26. 

45.00 

1.25 

12. 

47.50 

•41| 

27. 

93.00 

1.50 

13. 

49.00 

.43| 

28. 

90.00 

1.87! 

14. 

57.00 

.50 

29. 

105.00 

2.50 

15. 

81.00 

.56! 

30. 

440.00 

3.33! 


53. To find the cost of goods sold by the 100, 1000, or ton. 


l. Find the cost of 168 lb. of feed @ $1.10 per cwt. 


$1.68 = cost at $1 per cwt. 

.17 = cost at .10 per cwt. 
$1.85 = cost at $1.10 per cwt. 

hundredweight, 
weight, the cost 


First point off 2 places in the 
number of pounds, thus, 168 lb. = 
1.68 cwt. At $1 per hundredweight, 
the cost would be $1.68. At $.10 per 
Hence, at $1.10 per hundred- 


the cost would be $.168 or $.17. 
would be the sum of $1.68 and $.17, or $1.85. 











30 


ALIQUOT PARTS 


2. At SI.87 per C. find the cost of 634 bolts. 

634 = 6.34 C. 634 bolts = 6.34 hundreds. Since 1 

6 34 X SI 87 = $H 86 hundred costs $1.87, 6.34 hundreds cost 
6.34 times $1.87, or $11.86. 


3. At $15 per thousand, find the cost of 648 ft. of lumber. 
$6.48 = cost at 1 ^ per foot. $15 per M = l\£ per foot. At 

3.24 = cost at * i per foot. 1 * a £oot > 648 ft - cost ® 6 - 48 - and 

- at a foot, the cost is \ of $6.48, 

$9.72 = cost at $15 per M. or $ 3 . 24 . Hence, at a foot 

or $15 per M, the cost is $6.48 
+ $3.24, or $9.72. 

4. At $16.40 per M, find the cost of 13,400 bricks. 

13400 = 13.4 thousands. The number of M is found by 
13 4 X $16 40 = $219 76 Panting off 3 places in the number 
13,400, giving 13.4 M. 

At $16.40 per M, 13.4 M cost 13.4 
X $16.40, or $219.76. 

5. $13.50 per ton = ($6 + | of $6) per 1000 lb. 

Hence 2780 lb. at $13.50 per ton = 

$16.68 at $6 per 1000 lb. 

2.09 at $.75 per 1000 lb. 

$18.77 at $6.75 per 1000 lb., or $13.50 per ton. 


6 . $1.35 per 100 lb. = (1 1 + } i 


Hence 580 lb. at $1.35 per 100 lb. = 


+ To D per pound. 

$5.80 at 1 i per pound 
1.45 at J per pound 
.58 at to i per pound 
$7.83 at $1.35 per 100 lb. 


7. $31.25 per 1000 lb. = ($25 + \ of $25) per 1000 lb. 
Hence 6720 lb. at $31.25 per 1000 lb. = 

$168.00 at $25 per 1000 lb. 

42.00 at $6.25 per 1000 lb. 

$210.00 at $31.25 per 1000 lb. 


54. Devise a short method of finding the value of: 

8. 4370 lb. coal at $6.75 per ton. 

9. 5860 lb. hay at $15.75 per ton. 

10 . 22,470 brick at $11.25 per 1000. 






ALIQUOT PARTS 


31 


11 . 528 lb. meal at $1.75 per 100 lb. 

12 . 3156 lb. feed at $27 per ton. 

13. 17,640 ft. lumber at $23.75 per 1000 ft. 

14. 13,260 ft. lumber at $33.75 per 1000 ft. 

55. Solve mentally, if possible, the following: 
Find the cost of: 


1. 

220 lb. bran 

@ $1.35 

per cwt. 

@$1.40 

per cwt. 

2. 

350 lb. feed 

@ 2.50 

per cwt. 

@ 

2.25 

per cwt. 

3. 

675 lb. nails 

@ 5.50 

per cwt. 

@ 

5.75 

per cwt. 

4. 

875 lb. salt 

@ .80 

per cwt. 

@ 

.88 

per cwt. 

5. 

950 posts 

@ 6.25 

per C. 

@ 

6.75 

per C. 

6. 

1100 posts 

@ 7.25 

per C. 

@ 

7.70 

per C. 

7. 

1250 posts 

@ 8.00 

per C. 

@ 

7.20 

per C. 

8. 

15501b. lead 

@ 7.50 

per cwt. 

@ 

6.37 \ per cwt. 

9. 

38,4401b. pig iron @ 1.50 

per cwt. 

@ 

1.62J per cwt. 

10. 

24,6801b. sugar 

@ 5.50 

per cwt. 

@ 

5.75 

per cwt. 

11. 

1875 ft. lumber @35.00 

per M. 

@ 

45.00 

perM. 

12. 

1450 ft. lumber @ 25.00 

per M. 

@ 

22.00 

per M. 

13. 

23,360 brick 

@ 18.75 

per M. 

@ 

17.50 

per M. 

14. 

14,500 shingles 

@ 5.62J per M. 

@ 

5.25 

per M. 

15. 

346 ft. lumber @ 33.00 

per M. 

@ 

31.25 

per M. 

16. 

23,465 ft. lumber @ 37.50 

per M. 

@ 

33.75 

per M. 

17. 

47,860 brick 

@ 11.25 

per M. 

@ 

12.50 

per M. 

18. 

12,375 brick 

@ 13.75 

per M. 

@ 

15.00 

per M. 

19. 

7850 shingles 

@ 6.75 

per M. 

@ 

7.50 

per M. 

20. 

319 ft. lumber 

@ 45.00 

per M. 

@ 

55.00 

per M. 

21. 

2660 lb. hay 

@ 18.00 

per T. 

@ 

22.50 

per T. 

22. 

3474 lb. hay 

@26.00 

per T. 

@ 

24.00 

per T. 

23. 

3096 lb. straw 

@ 14.50 

per T. 

@ 

13.75 

per T. 

24. 

7276 lb. straw 

@ 13.75 

per T. 

@ 

12.50 

per T. 

25. 

1680 lb. feed 

@ 24.60 

per T. 

@ 

27.90 

per T. 

26. 

4288 lb. feed 

@ 26.00 

per T. 

@ 

27.00 

per T. 

27. 

13,890 lb. coal 

@ 13.75 

per T. 

@ 

12.50 

per T. 

28. 

47,976 lb. coal 

@ 13.50 

per T. 

@ 

11.25 

per T. 

29. 

1890 lb.coal 

@ 9.50 

per T. 

@ 

8.80 

per T. 

30. 

560 lb.coal 

@ 12.50 

perT. 

@ 

13.50 

per T. 


32 


ALIQUOT PARTS 
SIMPLE INTEREST 


ALIQUOT PARTS APPLIED TO SIMPLE INTEREST 

56. Simple interest is treated in this part of the text for two 
reasons: first, because it is required early in the course by all students 
who are at the same time pursuing the study of bookkeeping, 
and second, to correlate it with the work in “Aliquot Parts.” 

57. Interest is the price paid for the use of money. 

58. The principal is the sum for the use of which interest is paid. 

59. For ordinary interest calculations, a year is considered as 
having 12 mo. of 30 da. each, or 360 da. 

The expression “6% interest” or “interest at 6%” means, 

The interest on $1 for 1 year is $.06. 2 months is i of a year, 

therefore, 

The interest on $1 for 2 months (60 days) is $.01. Since the 
interest on $1 at 6% for 60 days is $.01, 

on $5 it is $.05 on $175 it is $1.73 

on $40 it is $.40 on $1000 it is $10.00, etc. 

Hence, to find the interest for 2 mo. or 60 days, at 6%, on any 
sum of money, move the decimal point TWO places to the left. 

60. Find the interest on $450.50 for 60 da. at 6%. 

$4|50.50 = $4.51, int. at 6% for 60 da. 

Since at 6% for 60 da. the interest on $1 is $.01, the interest on $450.50 is 
$.01 for each dollar, or $4.5050, which equals $4.51. (Answers in interest 
problems are always given in the nearest whole cent.) 

61. Find the interest at 6% on each of the following for 60 da.: 

1. $630 4. $846 7. $2465 10 . $824.75 

2. $750 5. $932 8. $3675 li. $968.66 

3. $365 6. $1036 9. $4270 12. $732.45 

62. In the preceding pages on aliquot parts 100 was the basis of 
comparison. In interest solutions 60 is the basis. Memorize the 
following aliquot parts of 60. (Compare with the parts of an 
hour.) 

30 = i of 60 12 = i of 60 5 = ^ of 60 

20 = i of 60 10 = i of 60 4 = T V of 60 

15 = \ of 60 6 = yu of 60 3 = of 60 


SIMPLE INTEREST 


33 


63. What combinations of 60 da. and aliquot parts of 60 da. 
will best give (a) 45 da.? (6) 27 da.? (c) 85 da.? 

(a) 60 da. (b) 60 da. (c) 60 da. 

15 da. = i of 60 20 da. = J of 60 20 da. = £ of 60 

45 da. 6 da. = X V of 60 _5 da. = } of 20 

1 da. = £ of 6 85 da. 

27 da. 


64. Find combinations of 60 da. and aliquot parts of 60 da. that 
will best amount to.: 


1. 55 da. 

2. 40 da. 

3. 33 da. 

4. 35 da. 

5. 25 da. 

6. 27 da. 

7. 32 da. 


8. 26 da. 

9. 66 da. 

10. 75 da. 

11. 72 da. 

12. 80 da. 

13. 87 da. 

14. 93 da. 


15. 84 da. 

16. 90 da. 

17. 11 da. 

18. 14 da. 

19. 16 da. 

20. 19 da. 

21. 21 da. 


22 . 76 da. 

23. 96 da. 

24. 105 da. 

25. 110 da. 

26. 135 da. 

27. 116 da. 

28. 101 da. 


65. l. Find the interest at 6% on $540 for 22 da. 


$5 

1 

~1 


40 = int. for 60 da. 
80 = int. for 20 da. 
18 = int. for 2 da. 
98 = int. for 22 da. 


22 da. is composed of the aliquot parts 
20 da. and 2 da. 

$5.40 is the interest for 60 da. 

20 da. is 1 of 60 da., hence the interest 
for 20 da. is ^ of $5.40, or $1.80. The remain¬ 


ing 2 da. is tV of 20 da., and the interest for 2 da. is T V of $1.80, or $.18. 
The interest for 22 da. is $1.98. 


2. Find the interest at 6% on $1716.75 for 87 da. 


$17 

1675 

= int. for 

60 

da. 

5 

7225 

= int. for 

20 

da. 

1 

7167 

= int. for 

6 

da. 


2861 

= int. for 

1 

da. 

$24 

8928 

= int. for 

87 

da. 

$24 

89 

= int. for 

87 

da. 


87 da. = 60 da. 4- 20 da. + 6 da. + 
1 da. Find 60 da. interest, which is 
$17.1675 and add to it £ of itself (for 
20 da.), tV of itself (for 6 da.), and & 
of the interest for 6 da. (for 1 da.) which 
gives $24.89, the interest for 87 da. 

Notes. (1) The vertical line takes 
the place of the decimal points in the 
solution. 


(2) The interest for fractional parts of the time should be kept correct to 
the 4th decimal place to insure accuracy in the cents column. 

VAN TUYL’S NEW COMP. AR.—3 






34 


ALIQUOT PARTS 


66. Find the i: 

nter 

l. 

$260 

for 

30 

da. 

2. 

$360 

for 

45 

da. 

3. 

$480 

for 

50 

da. 

4. 

$560 

for 

36 

da. 

5. 

$840 

for 

26 

da. 

6. 

$720 

for 

25 

da. 

7. 

$960 

for 

40 

da. 

8. 

$900 

for 

44 

da. 

9. 

$1260 

' for 

70 

da. 

10. 

$1600 

for 

90 

da. 

11. 

$1800 

for 

80 

da. 

12. 

$2000 

for 

75 

da. 


it at 6% on: 

13. $2100 for 72 da. 

14. $2200 for 96 da. 

15. $2600 for 105 da. 

16. $3000 for 16 da. 

17. $4200 for 30 da. 

18. $4800 for 33 da. 

19. $5100 for 45 da. 

20. $8000 for 45 da. 

21. $9000 for 50 da. 

22. $9600 for 90 da. 

23. $1000 for 75 da. 
42. $1340 for 120 da. 


25. $1600 for 150 da. 

26. $1800 for 210 da. 

27. $3000 for 180 da. 

28. $4000 for 240 da. 

29. $4500 for 320 da. 

30. $4800 for 160 da. 

31. $5000 for 150 da. 

32. $6300 for 160 da. 

33. $6600 for 140 da. 

34. $7200 for 115 da. 

35. $7500 for 100 da. 

36. $8400 for 99 da. 


67. l. What is the interest on $100 at 6% for 6 da.? 


6 da. is what part of 60 da.? 

What is the interest on $100 at 6% for 60 da.? 

What is the interest on $100 at 6% for 6 da.? 

What is the quickest way of finding the interest on $100 at 6% for 6 da.? 

What is the quickest way of finding the interest for 6 da. at 6% on any sum 

of money? 


2 . 


460 


Find the interest on $1460 at6% for (a) 6 days; (b) 9 days. 


(a) 

interest for 6 da. 


$2 


(b) 

460 = interest for 6 da. 
73 = interest for 3 da. 


19 = interest for 9 da. 


(a) Move the decimal point three places to the left. The 
result is the interest at 6% for 6 days. 

(b) After finding the interest at 6% for 6 days, find the 
interest for 3 days by taking § of the interest for 6 days. 
Adding the interest for 6 days and the interest for 3 days 
gives the interest for 9 days. 

Having the interest for 6 da., how may the interest be found for 

1 da.? 4 da.? 8 da.? 18 da.? 

2 da.? 5 da.? 9 da.? 24 da.? 

3 da.? 7 da.? 10 da.? 36 da.? 





SIMPLE INTEREST 


35 


t 


68 . Find the interest at 6% on the following: 


1. $400 for 6 da. 

2. $800 for 6 da. 

3. $1000 for 9 da. 

4. $1200 for 8 da. 

5. $1400 for 18 da. 

6. $1500 for 7 da. 

7. $1800 for 24 da. 


1 . 

$420 for 18 da. 
$540 for 20 da. 
$640 for 21 da. 
$350 for 13 da. 
$432 for 15 da. 
$128 for 16 da. 


8. $1560 for 36 da. 

9. $1280 for 42 da. 

10. $2000 for 39 da. 

11. $1250 for 48 da. 

12. $1125 for 24 da. 

13. $1440 for 19 da. 


2 . 

$630 for 11 da. 
$750 for 9 da. 
$780 for 26 da. 
$960 for 36 da. 
$1200 for 49 da. 
$840 for 50 da. 


15. $1480 for 24 da. 

16. $1600 for 27 da. 

17. $2100 for 7 da. 

18. $2400 for 3 da. 

19. $3600 for 2 da. 

20. $4200 for 1 da. 

21. $9000 for 4 da. 


3. 

$1300 for 66 da. 
$1560 for 72 da. 
$1480 for 78 da. 
$1260 for 84 da. 
$1450 for 90 da. 
$1660 for 91 da. 


14. $1280 for 21 da. 
69. Find the total interest at 6% on: 


4. 

$1728 for 95 da. 
$1890 for 98 da. 
$1820 for 100 da. 
$1675 for 108 da. 
$1440 for 110 da. 
$1360 for 120 da. 


5. 

$800 for 31 da. 
$900 for 41 da. 
$1000 for 48 da. 
$460 for 27 da. 
$390 for 19 da. 
$260 for 16 da. 


6 . 

$320 for 24 da. 
$390 for 2 da. 
$410 for 12 da. 
$812 for 15 da. 
$915 for 20 da. 
$990 for 10 da. 


7. 

$828 for 45 da. 
$936 for 15 da. 
$832 for 75 da. 
$648 for 70 da. 
$444 for 80 da. 
$555 for 40 da. 


8. 

$736 for 105 da. 
$888 for 110 da. 
$960 for 115 da. 
$822 for 150 da. 
$440 for 130 da. 
$375 for 88 da. 


9. 

$482.75 for 12 da. 
$321.50 for 18 da. 
$428.48 for 15 da. 
$320.10 for 20 da. 
$516.74 for 24 da. 
$608.31 for 30 da. 



10. 



$597.30 

for 

36 

da. 

$321.47 

for 

40 

da. 

$218.38 

for 

42 

da. 

$712.80 

for 

6 da. 

$811.75 

for 

10 

da. 

$928.30 

for 

30 

da. 


13. 



$720.30 

for 

25 

da. 

$628.50 

for 

44 

da. 

$731.20 

for 

14 

da. 

$906.75 

for 

9 da. 

$887.63 

for 

12 

da. 

$674.70 

for 

13 

da. 

16. 



$329.60 

for 

32 

da. 

$572.40 

for 

38 

da. 

$635.70 

for 

52 

da. 

$596.75 

for 

17 

da. 

$831.50 

for 

12 

da. 

$938.75 

for 

31 

da. 

19. 



$700 for 

•57 

da 



$1100 for 83 da. 
$1300 for 96 da. 
$1700 for 47 da. 
$1600 for 81 da. 
$1400 for 102 da. 

22 . 

$830 for 6 da. 
$870 for 9 da. 
$910 for 12 da. 
$530 for 15 da. 
$780 for 45 da. 
$880 for 42 da. 


ALIQUOT PARTS 

11 . 

$731.45 for 13 da. 
$806.40 for 14 da. 
$318.25 for 11 da. 
$208.14 for 10 da. 
$327.38 for 16 da. 
$445.50 for 19 da. 

14. 

$475 for 26 da. 
$938 for 33 da. 
$840 for 17 da. 
$729 for 19 da. 
$325 for 39 da. 
$480 for 47 da. 

17. 

$800 for 37 da. 
$900 for 44 da. 
$1200 for 39 da. 
$1500 for 53 da. 
$1800 for 67 da. 
$2000 for 78 da. 


20. 


$521.60 for 

23 da. 

$387.60 for 

32 da. 

$573.20 for 

41 da. 

$753.30 for 

55 da. 

$839.70 for 

64 da. 

$931.75 for 

20 da. 

23. 


$380 for 46 

da. 

$780 for 40 

da. 

$190 for 57 

da. 

$350 for 51 

da. 

$870 for 18 

da. 

$880 for 24 

da. 


12 . 

$673.87 for 21 da. 
$832.49 for 22 da. 
$738.27 for 25 da. 
$387.39 for 29 da. 
$873.93 for 5 da. 
$783.79 for 3 da. 

15. 

$425 for 38 da. 
$576 for 67 da. 
$545 for 88 da. 
$750 for 112 da. 
$800 for 99 da. 
$235 for 81 da. 

18. 

$450 for 28 da. 
$480 for 21 da. 
$540 for 19 da. 
$660 for 23 da. 
$720 for 32 da. 
$840 for 40 da. 

21 . 

$220 for 12 da. 
$330 for 15 da. 
$560 for 18 da. 
$640 for 24 da. 
$720 for 30 da. 
$960 for 10 da. 

24. 

$540 for 82 da. 
$840 for 12 da. 
$670 for 91 da. 
$900 for 29 da. 
$1200 for 39 da. 
$1400 for 90 da. 


SIMPLE INTEREST 


37 


INTEREST DRILL CHART 
70. Use this chart as suggested on page 27. 
Find the interest at 6% on: 


$800 for 30 da. 
$900 for 20 da. 
$600 for 15 da. 
$700 for 30 da. 
$640 for 15 da. 
$720 for 20 da. 
$480 for 10 da. 
$540 for 6 da. 
$400 for 9 da. 
$500 for 12 da. 
$800 for 15 da. 
$720 for 20 da. 
$960 for 30 da. 
$920 for 20 da. 
$860 for 30 da. 
$700 for 60 da. 
$600 for 10 da. 
$500 for 15 da. 
$400 for 75 da. 
$500 for 90 da. 
$600 for 80 da. 
$400 for 90 da. 
$800 for 75 da. 
$900 for 80 da. 
$600 for 90 da. 
$440 for 90 da. 
$780 for 30 da. 
$860 for 30 da. 
$460 for 30 da. 
$560 for 15 da. 
$660 for 20 da. 
$680 for 15 da. 
$240 for 120 da, 


$360 for 15 da. 
$320 for 15 da. 
$150 for 30 da. 
$250 for 120 da. 
$300 for 30 da. 
$400 for 6 da. 
$600 for 1 da. 
$800 for 6 da. 
$420 for 20 da. 
$350 for 12 da. 
$250 for 12 da. 
$750 for 20 da. 
$450 for 20 da. 
$150 for 20 da. 
$550 for 12 da. 
$1200 for 15 da. 
$1000 for 90 da. 
$1400 for 90 da. 
$1500 for 80 da. 
$1200 for 80 da. 
$1200 for 70 da. 
$1500 for 72 da. 
$1800 for 30 da. 
$1800 for 80 da. 
$1600 for 15 da. 
$1600 for 75 da. 
$1800 for 90 da. 
$2000 for 60 da. 
$2200 for 30 da. 
$2400 for 15 da. 
$3600 for 20 da. 
$3400 for 30 da. 
$3600 for 80 da. 


$4400 for 15 da. 
$4200 for 10 da. 
$4300 for 6 da. 
$4700 for 12 da. 
$4500 for 20 da. 
$4600 for 30 da. 
$5100 for 20 da. 
$5200 for 15 da. 
$4400 for 90 da. 
$4500 for 80 da. 
$4200 for 70 da. 
$4400 for 120 da. 
$4000 for 80 da. 
$3000 for 240 da. 
$3800 for 30 da. 
$3300 for 10 da. 
$3400 for 15 da. 
$5400 for 10 da. 
$4750 for 6 da. 
$4900 for 12 da. 
$4700 for 12 da. 
$2500 for 12 da. 
$2600 for 15 da. 
$2800 for 45 da. 
$2400 for 50 da. 
$3500 for 72 da. 
$3600 for 55 da. 
$4800 for 55 da. 
$5600 for 30 da. 
$3500 for 12 da. 
$3700 for 30 da. 
$6600 for 15 da. 
$2600 for 18 da. 


ADDITION 


71. Addition is the process of finding one number or quantity 
equal in value to two or more other numbers or quantities. 

72. The several numbers or quantities to be added are called 

addends. 

73. Investigation shows that the most frequently used arith¬ 
metical operation is simple addition. It is important that it be 
done rapidly and accurately. 

The following suggestions will be found helpful. 

ORAL DRILL EXERCISES 

74. l. Count by 3’s from 1 to 100; from 2 to 101. 

2. Count by 4’s from 1 to 101: from 2 to 102; from 3 to 103. 

3. Count by 5’s beginning with 0; 1; 2; 3; 4, until 100 is 
passed. 

4. Count by 6’s beginning with 0; 1; 2; 3; 4: 5, until 100 is 
passed. 

5. Count by 7’s beginning with 0; 1; 2; 3; 4; 5; 6, until 100 
is passed. 

6. Count by 8’s beginning with 0; 1; 2; 3; 4; 5; 6; 7, until 
100 is passed. 

7. Count by 9’s beginning with 0; 1 ; 2; 3; 4; 5; 6; 7; 8, until 
100 is passed. 

8. Count by 10’s beginning with 0; 1; 2; 3; 4; 5; 6; 7; 8; 9, 
until 100 is passed. 

9. Count by 11’s beginning with 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 
10, until 100 is passed. 

10. Count by 12’s beginning with 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 
10; 11, until 100 is passed. 


38 


RAPID DRILL EXERCISES 


39 


RAPID DRILL EXERCISES 


75. Drill on the following until you can state the results im¬ 
mediately : 


9 

8 

7 

6 

9 

4 

5 

3 

7 

9 

1 

8 

4 

2 

1 

3 

5 

6 

1 

2 

2 

1 

1 

3 

3 

3 

5 

2 

4 

2 

7 

1 

5 

5 

6 

4 

4 

2 

5 

6 

“ 

— 

— 

— 

— 

— 

— 

— 

— 

— 

9 

8 

7 

6 

9 

9 

7 

8 

9 

6 

1 

2 

3 

4 

5 

2 

4 

4 

7 

3 

9 

8 

7 

6 

5 

8 

6 

6 

3 

7 


~ 

— 

— 

— 

— 

— 

— 

— 

— 

17 

18 

24 

39 

43 

67 

76 

43 

61 

55 

2 

3 

5 

2 

8 

4 

2 

2 

1 

3 

3 

3 

4 

6 

1 

4 

7 

4 

7 

5 

5 

6 

1 

2 

1 

2 

1 

4 

2 

2 

38 

47 

64 

89 

79 

97 

48 

53 

89 

62 

9 

8 

3 

4 

3 

6 

5 

1 

9 

8 

1 

2 

7 

6 

7 

4 

5 

9 

1 

2 

56 

67 

83 

47 

82 

69 

43 

67 

79 

88 

2 

3 

6 

3 

8 

1 

4 

2 

3 

4 

4 

2 

3 

4 

1 

4 

4 

6 

6 

1 

4 

5 

1 

3 

1 

5 

2 

2 

1 

5 

81 

47 

74 

82 

71 

89 

74 

90 

27 

44 

6 

3 

4 

4 

6 

2 

6 

8 

3 

4 

2 

4 

5 

3 

1 

2 

1 

1 

4 

4 

2 

3 

1 

3 

3 

6 

3 

1 

3 

2 

76. 

Two important aids to accuracy and speed in addition work 

are: (1) to make the figures neatly and of uniform 
write them exactly in columns. 

Add: 

size 

; and (2) to 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

64 

59 

94 

56 

49 

59 

67 

61 

67 

56 

28 

63 

89 

38 

84 

63 

38 

49 

84 

73 

39 

87 

21 

27 

47 

72 

79 

72 

98 

87 







40 ADDITION 


11. 

12. 

13 . 

14 . 

15 . 

16 . 

17 . 

18. 

19. 

20 . 

69 

19 

64 

56 

59 

53 

48 

53 

43 

53 

76 

66 

83 

79 

32 

87 

72 

89 

72 

87 

87 

74 

39 

96 

27 

98 

28 

72 

29 

91 

24 

47 

77 

87 

87 

82 

63 

68 

38 

76 

21. 

22. 

23. 

24. 

25. 

26. 

27. 

28. 

29. 

30 . 

43 

74 

92 

94 

69 

88 

63 

48 

56 

45 

82 

91 

28 

76 

78 

72 

79 

27 

74 

97 

79 

43 

74 

87 

86 

38 

64 

68 

83 

86 

87 

87 

68 

92 

97 

76 

87 

93 

92 

73 

64 

96 

93 

47 

74 

43 

99 

27 

76 

49 

47 

78 

72 

83 

83 

89 

76 

64 

57 

36 

82 

47 

46 

97 

72 

96 

64 

56 

83 

17 

96 

64 

87 

48 

87 

74 

87 

74 

92 

85 


77. The rapidity of addition may be increased by adding figures 
by groups. Note the groups of 10 in the following. Add rapidly: 


l. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

4 

7 

4 

6 

3 

9 

7 

4 

5 

6 

6 

3 

3 

3 

4 

1 

3 

4 

3 

2 

8 

9 

3 

1 

3 

4 

6 

2 

2 

2 

2 

1 

7 

4 

5 

2 

2 

5 

4 

3 

7 

5 

2 

4 

2 

3 

2 

4 

2 

5 

3 

5 

1 

2 

3 

1 

8 

1 

4 

2 



~ 






— 

— 

li. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

37 

49 

67 

56 

56 

47 

31 

77 

47 

49 

84 

36 

84 

34 

74 

39 

63 

69 

83 

84 

97 

83 

97 

73 

64 

28 

39 

83 

76 

39 

26 

92 

32 

91 

59 

37 

87 

65 

64 

73 

84 

78 

43 

79 

76 

73 

79 

27 

96 

42 

97 

46 

88 

65 

85 

81 

73 

39 

89 

87 

43 

87 

79 

82 

93 

79 

47 

47 

97 

65 




CHECKING THE WORK 


41 


CHECKING THE WORK 

78. Checking is a means of testing the accuracy of the work. 
Accuracy can be obtained by frequently checking the results. 

79. The best method of checking addition is to add the columns 
in the reverse order. If the column is first added from the bottom 
up, check by adding from the top down. 

80. Many bookkeepers and accountants use the method shown 
in the following illustrative solution: 


Add: 

(1st) 

26 

31 

32 

27 


3967 

8234 

7987 

4372 

5976 


(Check) 

27 

32 

31 

26 


30536 30536 30536 


By adding the units’ column, a sum 
of 26 is obtained, which is written as 
shown. Next, by adding the tens’ 
column, the sum is found to be 31, 
which is written as in the illustration. 
The sums of the hundreds’ and thou¬ 
sands’ columns are 32 and 27, re¬ 
spectively, which are written as here 
shown. 


147=12= 3 
229 =13 = 4 
331 = 7 = 7 


707 


Beginning with the thousands’ column and adding in the opposite direction 
we find the sum to be 27, which is written as shown in the column marked 
“Check.” The sum of the hundreds’, tens’, and units’ columns, which are 32, 
31, and 26, respectively, are written as in the illustration. By adding these 
several sums the result is seen to agree with the first result obtained. Hence 
the work is probably correct. 

81. Casting out nines. 

The sum of the digits 1, 4, and 7 is 12, equal to one 9 
and 3 over. The sum of the digits in the second addend is 
13, equal to one 9 and 4 over; and the sum in the third 
addend is 7, equal to no 9’s and 7 over. The sum of the 
several remainders, 3, 4, and 7, is 14, equal to one 9 and 
5 over. The sum of the digits in the sum, 707, is 14, equal 
to one 9 and 5 over. Hence the addition is probably 
correct. 

Notes. (1) The digits of 147 are the separate numbers 1, 4, and 7. 

(2) The remainders 3, 4, 7, and 5 are the sums of the digits in 12, 13, 7, and 
14, respectively. 12 is equal to one 9 with a remainder of 3. The division is 
unnecessary, however, as the excess (remainder) can always be found by 
adding the digits. 

(3) The remainders 3, 4, and 7 are generally called the “excess of nines,” 
and the principle is stated thus: “The excess of nines in a sum is equal to the 
excess in the sum of the excesses.” 

(4) Casting out nines is not an absolute test of accuracy. Transposition of 
figures cannot be detected by this check, nor can the omission or addition of 
9’s or ciphers be discovered. 


14 =14 
5= 5 






42 


ADDITION 


82. Casting out elevens. 


475 =2 
382 =8 
479 =6 
1336 =5 


Find the sum of the digits in the odd places (beginning at 
units) and from it deduct the sum of the digits in the even 
places. If the sum of the digits in the even places exceeds the 
sum of those in the odd places, add 11, or some multiple of 11, 
to the sum of the digits in the odd places. 


Thus, 5+4-7 =2 

2+3 + 11 —8 = 81 6 + 8 + 2 = 16; 
9+4—7 = 6 16 = one 11 and 5 over. 

6+3 -3 - 1=5. 


Note. Casting out elevens will detect errors not located by the nine check, 
as indicated in note (4), p. 41. 


83. Add the following, and check the results: 


1 . 

2. 

3. 

4. 

5. 

6. 

2874 

8727 

8237 

9371 

6738 

3796 

3957 

3469 

6489 

6487 

9674 

3987 

6498 

5487 

5387 

5238 

5976 

4598 

7237 

3245 

6959 

9764 

8238 

3294 

4596 

5894 

6432 

5437 

9783 

5786 

3872 

7978 

8643 

6087 

4598 

5487 

5295 

5673 

9782 

9121 

1359 

7372 

9638 

9178 

9176 

4327 

9385 

9765 


84. Each of the following columns should be added in fifteen 
seconds. Drill on them until correct results can be obtained in 
that time. 


l. 

2. 

3. 

4. 

5. 

6. 

4789 

7968 

4894 

4893 

4389 

1531 

3847 

7324 

3762 

3469 

5767 

4574 

5963 

8759 

5437 

9276 

9247 

8276 

8769 

4365 

9864 

1498 

6389 

9832 

5271 

9859 

4767 

9763 

9127 

4976 

8728 

7312 

5943 

7987 

8784 

2872 

9592 

8764 

8976 

6477 

4728 

3193 

7352 

5231 

4382 

9816 

2854 

8715 

5897 

9827 

4767 

8125 

1645 

2810 

7684 

5876 

5723 

7784 

5196 

4567 



















ADDITION 


43 

7. 

8. 

9. 

10. 

li. 

32478672 

42876597 

45967248 

12345678 

24681357 

45972861 

27638976 

53876289 

98765432 

91352468 

56972764 

45972489 

93856214 

87654321 

57913546 

36824579 

72510764 

72387276 

34567892 

94872468 

59472876 

63897123 

10468217 

45678923 

74692789 

12. 

13. 

14 

15. 

16. 

76945897 

54763247 

57689727 

47897684 

98753214 

34872469 

98674382 

68972543 

32076483 

76543216 

89778976 

76948269 

74202604 

82478978 

97531246 

69877978 

63897476 

70876437 

68764376 

89765382 

45987678 

48769531 

97687248 

98765432 

74974768 

85. Each of the following columns should be added in one minute 

Drill on them until it can be done with facility and accuracy. 


2. 

3. 

4. 

5. 

1 . 34568 

93876 

69784 

38976 

94768 

2. 72487 

47876 

72864 

72487 

48694 

3. 95438 

83297 

14897 

92764 

38972 

4. 64876 

45872 

32571 

12476 

66666 

5. 45987 

76978 

89724 

53897 

89764 

6 . 98321 

32589 

89724 

69748 

92787 

7. 87643 

13489 

97247 

48724 

77777 

8 . 97987 

87943 

83971 

93697 

48938 

9. 34786 

76897 

72481 

54328 

55555 

10 . 49876 

37643 

92482 

97648 

48743 

11 . 24689 

28971 

76893 

63872 

33333 

12 . 39872 

59761 

64762 

59763 

98697 

13 . 97246 

97246 

59769 

97687 

88888 

14. 72489 

38972 

43872 

53792 

46479 

15. 93247 

89726 

74672 

47867 

99999 

16. 59876 

97642 

94897 

83897 

72687 

17 . 72489 

57921 

58932 

46876 

22222 

18. 72979 

38761 

72468 

93872 

48769 


Suggestion. The number of problems in the above columns can 
be increased by requiring pupils to find the sum of numbers 1 to 10 
inclusive; 2 to 11; 3 to 12; 4 to 13; etc. 

















44 


ADDITION 


6. The following is a list of the twenty largest cities in the 
United States in order of size, according to the last census. Find 
their total population. New York, N. Y., 5,620,048; Chicago, 
Ill., 2,701,705; Philadelphia, Pa., 1,823,779- Detroit, Mich., 993,678; 
Cleveland, O., 796,841; St. Louis, Mo., 772,897; Boston, Mass., 
748,060; Baltimore. Md., 733,826; Pittsburgh, Pa., 588,343; Los 
Angeles, Cal., 576,673; Buffalo, N. Y., 506,775; San Francisco, 
Cal., 506,676; Milwaukee, Wis., 457,147; Washington, D. C., 
437,571; Newark, N. J., 414,524; Cincinnati, O., 401,247; New 
Orleans, La., 387,219; Minneapolis, Minn., 380,582; Kansas City, 
Mo., 324,410; Seattle, Wash., 315,312. 

86. Problems like the following will be interesting and helpful 
for rapid addition drills: 

32347 The teacher dictates two numbers of five figures each, 

46943 the student writing them as shown in the margin. 


The remaining numbers in the column are found by 
the student in this manner: 

The third number is the sum of the first two numbers; the 
fourth number is the sum of the second and third numbers; the 
fifth number is the sum of the third and fourth numbers; etc. 
When there are ten numbers the whole column is added. As the 
column is written, every third number is checked, as here shown, 
so as to tell quickly when the tenth number is written. With a 
little practice, students should find the final total in one minute 
and thirty seconds from the time the second number is dictated. 

5910069 


32347 

46943 

79290V 

126233 

205523 

331756V 

537279 

869035 

1406314V 

2275349 


Extend columns to ten numbers and add: 

1.47568 4.29384 7.74632 

10. 35798 

56473 

75867 

96879 

98765 

2. 65748 

5. 73645 

8. 85769 

ll. 12345 

65342 

92834 

87965 

67892 

3. 82736 

6. 29384 

9. 84754 

12. 98647 

54637 

94857 

85764 

86793 

















HORIZONTAL 

ADDITION 

45 

13. 57463 
65374 

21. 94857 

73645 

29. 37465 

43265 

37. 74635 
95864 

14. 21364 
56784 

22. 73645 

12385 

30. 48576 

19867 

38. 53426 
63545 

15. 39485 
23876 

23. 92836 

84756 

31. 95843 

41567 

38. 47586 
75648 

16. 39264 
23876 

24. 82734 

93754 

32. 38475 

85769 

40. 29731 
63549 

17. 36482 
84756 

25. 12938 

91542 

33. 64573 

75643 

41. 75312 
21357 

18. 12935 
94857 

26. 45678 

64532 

34. 43789 . 

59764 

42. 15486 
35943 

19. 39486 
21&5 

27. 67891 

45768 

35. 97856 

89543 

43. 52565 
43464 

20. 39485 
31495 

28. 38475 

93847 

36. 34768 

56789 

44. 98991 
75728 


HORIZONTAL ADDITION 

87. Addition from left to right is important on invoices, time 
sheets, sales sheets, statistical reports, etc. 

88. Add from left to right. Check by adding from right to left. 

1. 4, 8, 7, 6, 3, 2, 7, 9. 

2. 7, 9, 6, 4, 5, 8, 9, 7. 

3. 6, 4, 7, 9, 8, 7, 9, 5, 6. 

4. 15, 18, 29, 73, 96, 47, 85, 93, 88. 

5. 57, 93, 84, 96, 82, 78, 49, 56, 97. 

6. 72, 89, 37, 43, 87, 92, 46, 34, 73. 

7. 67, 83, 96, 38, 72, 47, 93, 89, 92. 

































46 


ADDITION 


Add the following by columns and from left to right. Check the 
result by adding the horizontal and vertical totals. 


8 . 

54 + 78+72 = 
36 + 29+83 = 
85 + 34+96 = 
92 + 56+87 = 

+ + = 

11 . 

382 + 592+848 = 
961 + 796+796 = 
729 + 816+382 = 
384 + 723+794 = 
+ + = 


9. 

38 + 28+54 = 
27 + 36+38 = 
54 + 79+29 = 
93 + 87+98 = 

+ + = 

12 . 

568 + 827+584 = 
729 + 968+763 = 
384 + 817+972 = 
754 + 927+475 = 
+ + = 


10 . 

74 + 37+45 = 
56 + 78+58 = 
89+56 +79 = 
79 + 33+46 = 
+ + = 

13. 

382 + 916+538 = 
976 + 693+872 = 
438 + 821+279 + 
728 + 572+186 = 
+ + = 


14. Find the total sales in all departments for each month, and 
the total sales in each department for the year. Check the work 
by adding the two sets of totals. 


Summary of Monthly Sales 



Clothing 

Dress Goods 

Gloves 

Hats 

Shoes 

Totals 

Jan. . . 

$16,427.85 

$12,427.83 

$3,472.50 

$5,472.60 

$7,592.50 


Feb. . . 

15,138.20 

11,428.50 

2,896.75 

4,854.30 

6,438.90 


Mar. . . 

18,278.26 

12,736.25 

3,124.60 

4,472.60 

5,372.50 


Apr. . . 

17,493.87 

10,964.78 

2,148.75 

4,134.50 

5,796.25 


May . . 

15,382.97 

9,974.63 

2,004.50 

3,834.75 

5,176.85 


June . . 

14,582.76 

9,864.59 

1,995.70 

3,564.75 

6,487.50 


July . . 

13,996.80 

9,645.72 

1,875.50 

3,472.25 

6,193.85 


Aug. . . 

14,592.73 

8,796.21 

1,850.60 

3,678.50 

5,040.45 


Sept. . . 

15,827.64 

9,793.13 

1,924.30 

4,075.65 

6,384.90 


Oct. . . 

18,973.90 

12,897.08 

2,394.60 

4,763.83 

6,876.75 


Nov. . . 

22,738.75 

14,594.77 

3,548.75 

5,276.90 

7,564.55 


Dec. . . 

28,693.72 

18,798.99 

3,875.50 

5,798.75 

8,097.55 


Totals . 

l 






















HORIZONTAL ADDITION 


47 


Give the horizontal, vertical and grand totals of the following: 


Sales 

Uniforms 

Leggings 

Caps 

Totals 

15. Monday . . . 

$1249.96 

$28.75 

$368.42 


Tuesday . . . 

1585.98 

392.52 

793.65 


Wednesday . . 

1896.47 

87.50 

715.26 


Thursday . . 

807.96 

92.75 

156.91 


Friday .... 

1193.53 

136.50 

653.65 


Saturday . . 

3168.49 

148.25 

876.52 


Totals . . . 






Sales 

Dress Goods 

Shoes 

Notions 

Totals 

16. Monday . . . 

$87.68 

$18.75 

$16.83 


Tuesday . . . 

125.76 

26.82 

9.76 


Wednesday 

235.67 

76.85 

23.82 


Thursday . . 

89.62 

89.75 

48.98 


Friday .... 

123.98 

110.60 

75.67 


Saturday . . 

982.76 

234.50 

163.97 


Totals . . . 






Months 

Books 

Shoes 

Millinery 

Dry Goods 

Totals 

17. July . . . 
August . . 
September . 
October . . 
November . 
December . 

Totals . 

$4653.90 

3253.76 

4324.45 
8780.19 
3738.98 

7544.45 

$6543.87 

4324.80 

7342.98 

8756.65 

3655.45 

7298.90 

$5346.75 

7636.65 

4898.90 

3756.98 

8765.89 

9755.35 

$8235.53 

9845.75 

7342.89 

6547.87 

8453.00 

9879.00 








Postal Revenue at One Post Office 



First 

Class 

Second 

Class 

Third 

Class 

Parcel 

Post 

Special 

Deliv¬ 

ery 

Regis¬ 
try • 

Insur¬ 

ance 

Totals 

18. M. . . 

$106.22 

$23.15 

$13.24 

$43.41 

$3.60 

$4.50 

$6.80 


T. .. 

145.61 

33.43 

21.13 

57.57 

4.50 

5.30 

7.75 


W. . . 

131.99 

19.22 

14.56 

49.92 

3.10 

1.30 

4.80 


Th. . . 

201.13 

31.52 

22.10 

28.88 

4.70 

3.90 

1.50 


F. 

187.67 

20.78 

31.57 

65.72 

7.70 

7.50 

2.50 


S. .. 

149.78 

31.56 

43.44 

38.48 

4.80 

6.50 

3.45 


Totals 




























































48 


ADDITION 



18. In the following Abstract Sales Book find (1) the total 
daily sales by adding the sales on account and the sales for cash; 
(2) the total weekly sales by adding the total daily sales for each 
week; and (3) the total monthly sales by adding the several weekly 
totals. 


19. 


ABSTRACT SALES BOOK 


Date 


From Sales 
Sheet No. 

Sales On 
Account 

Sales for 
Cash 

Total 

Daily 

Aug. 

1 

122 

132 

16 

87 

85 




2 

123 

278 

32 

115 

64 




3 

124 

225 

83 

79 

69 




4 

125 

196 

93 

107 

86 




5 

126 

209 

87 

45 

90 




6 

127 

367 

64 

128 

72 

— 



8 

128 

119 

81 

58 

27 




9 

129 

109 

54 

69 

27 




10 

130 

99 

82 

72 

33 




11 

131 

116 

93 

81 

57 




12 

132 

122 

11 

76 

22 




13 

133 

295 

60 

117 

09 

— 

— 


15 

134 

98 

90 

63 

76 




16 

135 

113 

13 

71 

21 




17 

136 

119 

17 

69 

80 




18 

137 

97 

55 

92 

96 




19 

138 

164 

73 

97 

85 




20 

139 

315 

76 

118 

24 


— 


22 

140 

98 

76 

72 

13 




23 

141 

101 

27 

78 

17 




24 

142 

111 

12 

91 

16 




25 

143 

127 

31 

105 

16 




26 

144 

172 

85 

116 

23 




27 

145 

417 

81 

193 

82 

— 

— 


29 

146 

107 

88 

69 

28 




30 

147 

163 

91 

121 

11 




31 

148 

136 

54 

99 

83 













Total 

Weekly 


Total 

Monthly 

































ADDITION DRILL CHART 


49 


Addition drill chart. 

89. This chart is intended to help the pupil learn to add num¬ 
bers by groups of two and three figures each. The light 
lines are to divide the columns into such groups of figures. For 
class drill these or similar columns should be written on large 
sheets of paper so that they may be read across the room. As 
the teacher points to any group the pupils will state the sum at 
sight. Then an individual column should be added two figures at 
a time. Drill in horizontal addition may be had here also. 

The teacher will easily discover many ways of using this chart. 
It should be used frequently. 


82 93 84 76 583 928 


92 

19 

12 

12 

29 

29 

19 

93 
92 

29 

92 

93 
29 

38 

93 

93 

93 

92 


83 74 


938472 


738 
746 

374 
983 

284 
847 

485 
341 

472 
485 

384 
928 

849 
948 
383 
476 
65 839 284 


384 

283 

728 

263 

341 

493 

736 

829 

938 

839 

829 

583 

465 

849 

829 

648 


93 84 72 93 847 563 


92 83 


1283 74 65 537283 


83 928 


394 

364 

374 

293 

392 

394 

394 

638 

583 

398 

848 

384 


64 849 
64 839 
638 
726 
472 


64 839485 


361 

872 

827 

625 

847 

834 

857 

857 

495 

928 

468 

374 

746 

384 

485 

291 

384 

834 


39 

48 

57 

392 

837 

38 

47 

38 

293 

847 

38 

47 

28 

394 

872 

38 

47 

26 

272 

534 

83 

94 

82 

736 

482 

29 

38 

47 

283 

949 

74 

29 

38 

472 

634 

74 

63 

84 

958 

374 

84 

95 

93 

847 

562 

39 

48 

57 

362 

635 

37 

49 

28 

374 

628 

84 

75 

13 

849 

599 

38 

47 

46 

382 

937 

38 

47 

46 

273 

846 

57 

94 

85 

748 

598 

94 

87 

56 

384 

791 

84 

75 

62 

837 

469 

83 

72 

63 

948 

728 

84 

75 

69 

384 

756 

37 

49 

28 

374 

637 


VAN TUYL’S NEW COMP. AR.- 



















SUBTRACTION 


90 . Subtraction is the process of finding a number or quantity 
equal to the difference between two other numbers or quantities. 

91 In expressions involving a parenthesis (), as 16 — (5+7), the 
operations indicated within the parenthesis must be performed 
before the operation indicated by the sign before the parenthesis. 
Thus, in the example given, 5 and 7 must be added first, and their 
sum, 12, taken from 16. 

The vinculum-has the same effect as the parenthesis. 

Thus, 16-5+7 = 4. 

ORAL DRILL EXERCISE 

92 . l. Count backwards by 2’s beginning with 100; 99. 

2. Count backwards by 3’s beginning with 100; 99; 98. 

3. Count backwards by 4’s beginning with 100; 99; 98; 97. 

4. Count backwards by 5's beginning with 100; 99; 98; 97; 96. 

5. Count backwards by 6’s beginning with 100; 99; 98; 97; 
96; 95. 

6. Count backwards by 7’s beginning with 100; 99; 98; 97;96; 
95; 94. 

7. Count backwards by 8’s beginning with 100 ; 99 ; 98 ; 97 ; 96 ; 
95; 94; 93. 

8. Count backwards by 9’s beginning with 100; 99; 98; 97; 96; 
95; 94; 93; 92. 

9. Count backwards by 10’s beginning with 100; 99; 98; 97; 
96; 95; 94; 93; 92; 91. 

10. Count backwards by ll’s beginning with 100; 99; 98; 97 ; 
96; 95; 94; 93; 92; 91; 90. 

11. Count backwards by 12’s beginning with 100* 99; 98; 97; 
96; 95; 94; 93; 92; 91; 90; 89. 


50 



RAPID DRILL EXERCISE 


51 


93. Subtract quickly: 


1 . 

2 . 

3 . 

4 . 

5 . 

6 . 

7 . 

8 . 

9 . 

10 . 

27 

23 

24 

27 

22 

23 

25 

27 

28 

26 

6 

8 

12 

9 

4 

7 

9 

8 

9 

7 






— 

— 




11 . 

12 . 

13 . 

14 . 

15 . 

16 . 

17 . 

18 . 

19 . 

20 . 

32 

35 

38 

32 

39 

37 

36 

34 

33 

31 

19 

17 

19 

17 

18 

18 

19 

17 

16 

15 

21 . 

22 . 

23 . 

24 . 

25 . 

26 . 

27 . 

28 . 

29 . 

30 . 

54 

59 

58 

54 

56 

52 

51 

57 

53 

55 

36 

32 

39 

37 

38 

39 

34 

38 

37 

38 


RAPID DRILL EXERCISE 

94. Subtract: 


l. 

2 . 

3 . 

4 . 

5 . 

6 . 

7 * 

890467 

740231 

947204 

943876 

740023 

630704 

845700 

387589 

674678 

678327 

427628 

287204 

470568 

687354 

8 . 

9 . 

10 . 

11 . 

12 . 

13 . 

14 . 

472907 

907240 

548724 

324876 

597860 

938701 

300470 

293879 

689382 

279876 

128987 

407976 

764902 

298784 

15 . 

16 . 

17 . 

18 . 

19 . 

20 . 

21 . 

487246 

697287 

320765 

596872 

729684 

864200 

975310 

398472 

472349 

162782 

372849 

537315 

795301 

682405 

22 . 

23 . 

24 . 

25 . 

26 . 

27 . 

28 . 

108724 

398764 

587639 

438076 

987654 

438762 

487296 

106897 

297876 

395301 

278649 

321076 

287964 

245498 

29 . 

30 . 

31 . 

32 . 

33 . 

34 . 

35 . 

568397 

497872 

495287 

863829 

397286 

327693 

978531 

298729 

387692 

278469 

438297 

276397 

278798 

299729 







































52 


SUBTRACTION 


CHECKING THE WORK 

95. The best method of checking the work in subtraction is to 
add the remainder to the subtrahend. If the sum is equal to the 
minuend, the work is correct. 

From 1476 take 938. 

1476 

938 

538 Check. 938+538 = 1476. 

Check the work on the preceding page. 

THE “MAKING CHANGE” OR AUSTRIAN METHOD 

96. This method of subtraction consists in adding to the amount 
of the purchase enough change to make the sum equal to the 
amount paid. 

1. A man buys groceries to the value of $1.38, and gives a two- 
dollar bill in payment. How much change should he receive? 

The cashier in making change will return to the customer 2 cents, a ten- 
cent piece, and a half dollar (saying, “$1.38, 40, 50, $2.00,” which means 
$1.38+$.02 = $1.40; $1.40+$.10 = $1.50; and $1.50+$.50 = $2.00). 

Of course, other coins than those mentioned might be returned by the 
cashier, but it is customary to make change in the largest coins possible. 

Note. This method is used by many in ordinary subtraction problems. 

2. From 1231 take 573. 

1231 

57^ 3+8 = 11; write 8. 

-- 7+1 (carried) +5 = 13; write 5. 

658 5 + 1 (carried)+6 = 12; write 6. 

97. What change should be given for a two-dollar bill if the 
following purchases were made? for a five-dollar bill? 


Name the coins or bills given as change. Use the largest de¬ 
nominations possible. 


1. 

$1.56 

8 . 

$1.17 

15 . 

$ .72 

22 . 

$1.82 

2 . 

1.27 

9. 

.96 

16 . 

.49 

23 . 

1.54 

3 . 

.87 

10 . 

.88 

17 . 

.27 

24 . 

1.43 

4 . 

.58 

11. 

1.49 

18. 

1.67 

25 . 

.40 

5 . 

1.33 

12. 

1.68 

19 . 

1.76 

26 . 

1.32 

6. 

1.64 

13 . 

1.21 

20. 

1.87 

27. 

1.41 

7. 

1.29 

14 . 

1.11 

21. 

1.19 

28 . 

.99 


MAKING CHANGE” OR AUSTRIAN METHOD 


53 


This method of subtraction has an important application in all 
problems in which the sum of several numbers is to be taken from 
another number. 

29 . A man deposits SI625 in a bank and draws checks against it 
for S122, $245, and $347. Find his balance. 


$1625 


122 

245 

347 

911 

7+9 = 16. 


7+5+2 = 14; 14+1 = 15. The units’ figure of 14 is 1 less 
than the units’ figure of the minuend; hence 1 is added to the 14 
to make its units’ figure equal to the units’ figure of the minuend. 
The 1 is written as the first or units’ figure of the difference. 
Carry 1 (the tens’ figure in 15) and add the tens’ column. 1 +4 
+4+2 = 11; 11+1=12. As before, write 1 in the remainder, 
and carry 1. Add the hundreds’ column. 1 +3+2+1 =7; 
Write 9 in the remainder. The bank balance, then, is $911. 


Find the balance of the following bank accounts: 


Total Deposits 


Checks Drawn 

30. $ 785.00 . 

... $ 76.00, 

$45.00. 


31. $ 427.50 . 

... $ 37.50, 

$25.00. 

$100.00. 

32. $1287.70 . 

. . . $128.50, 

$97.60, 

$ 52.70. 

33. $2576.28 . 

. . . $528.75, 

$ 7.50, 

$923.65. 

34. $3789.10 . 

. . . $227.50, 

$ 1.75, 

$ 85.00, 

35. Find the balance of the following account: 


Withdrawal J. McLOUGHLIN Investment 


Gain 



36 . Find the loss or gain on merchandise: 

































54 


SUBTRACTION 


37. Find the balance of cash: 



38. Find the loss: 



THE COMPLEMENT METHOD 


98. The complement of a number is the difference between 
that number and the unit of a next higher order. 

3 is the complement of 7, because 3 is the difference between 7 and 10; 31 is 
the complement of 69, because 31 is the difference between 69 and 100; etc. 

99. The complement method of subtraction is based on the 
principle illustrated in the following examples: 

1. From 23 take 8. 

23 - 8 = (23+2) - (8+2) = 25 -10 = 15. 

The complement of 8 is 2. 

If 2 is added to the minuend and is then deducted by adding it to the sub¬ 
trahend also, it is evident that the value of the remainder will be unchanged. 
Hence, in the above example we think 23, 25, 15. 

2. From 173 take 96. 

173-96 = (173+4) - (96+4) = 177 -100 = 77. 

The complement of 96 is 4. 

As stated in the preceding example, if 4 is added to both minuend and sub¬ 
trahend, the remainder is unchanged. Think 173, 177, 77. 

This method of subtraction is specially convenient when one number is to be 
deducted from the sum of several other numbers. 






























THE COMPLEMENT METHOD 


55 


From the sum of 827, 136, and 472, take 658. 


827 Arrange the numbers as shown. The complement of 658 is 342. 
136 (The complement should be carried in the mind, not written on the 
472 P a P er -) Add the complement to the several numbers of the minuend, 

- and deduct 100, (that is, 1 unit of the next higher order than the sub- 

658 trahend). The result is 777. 

777 


100. By the method just explained solve the following: 


1. 48+27+63-85 

2. 39+48+97-64 

3 . 149 + 164+387-456 

4 . 827+639+564-387 

5 . 964+763+816-978 

6. 839+572+634-857 


7 . 768 + 173+824-643 

8. 246+369+741-927 

9 . 329+417+726-843 

10. 274+976+834-748 

11. 1238+1376+2473-3764 

12. 3872+4391+7684-8727 


13. The following statement shows the earnings and the expenses 
of a certain railroad company for two months Find the total earn¬ 
ings and expenses; the increase or decrease of each class of earn¬ 
ings and expenses for November as compared with October; and 
the net increase or decrease. 



November 

October 

Increase 

Decrease 

Earnings 

Coal traffic 

Merchandise 

Passengers 

Express 

Mails 

Miscellaneous 

8850,456.25 

625,382.40 

227,421.36 

16,172.44 

11,328.38 

24,275.60 

$550,321.30 

582,498.22 

231,375.60 

17,275.85 

10,531.28 

25,386.24 



Total earnings 
Expenses 

Maintenance of road 
Maintenance of equipment 
Transportation 

General 

$150,381.35 

192,571.90 

602,892.70 

60,428.75 

$146,327.88 

207,364.76 

570,192.60 

63,175.80 



Total expenses 

Net earnings 
























56 


SUBTRACTION 


14 . In the following abstract purchase book find, (1) the amount 
of cash paid for each purchase, (2) the total amount of all invoices, 
the total amount of discount, and the total amount paid for 
merchandise for the month, (3) check the work by subtracting 
the total discount from the total amount of invoices to see if it 
agrees with the amount paid. 






























MULTIPLICATION 

101. Multiplication is the process of taking one number as 
many times as there are units in another. 

102. The multiplier and the multiplicand are called factors of 
the product 

103. To multiply by 10 or a multiple of 10. 

1. Multiply 4586 by 10, by 100, by 1000. 

4586 X 10 = 45860 
4586 X 100 = 458600 
4586X1000 = 4586000 

Hence, to multiply by 10, 100, 1000, etc., annex as many zeros to the 
multiplicant as there are zeros in the multiplier. 

2 . Multiply 345 by 30; by 400; by 5000. 

345 X 30 = 10350 
345 X 400 = 138000 
345X5000=1725000 

Hence, to multiply by 30, 400, 5000, etc., multiply the multiplicand by the 
3, 4, or 5, etc., and annex as many zeros to the product as there are zeros 
in the multiplier. 

3. Multiply 4300 by 6000. 

43X6 = 258; hence 25800000. 

Omit all zeros on the right of the factors, and multiply the remaining num¬ 
bers. To the result annex as many zeros as were omitted from both factors. 

104. Make the following multiplications mentally where possible: 

1. 472 by 10 3. 596 by 1000 5. 920 by 100 

2. 563 by 100 4. 738 by 10 6. 6472 by 1000 


57 


58 


SHORT METHODS 


7. 5345 by 10 

8. 824 by 300 

9. 9300 by 500 

10. 7250 by 6000 

11. 82,400 by 5000 


12. 93,875 by 100 

13. 92,700 by 2000 

14. 4300 by 700 

15. 89,600 by 4000 

16. 97,620 by 100 


16. 83,472 by 10 

18. 94,800 by 800 

19. 7000 by 5000 

20 . 8900 by 900 

21 . 9800 by 3000 


105. To multiply by 25, 50, 75, etc. 

1. Multiply 413 by 25; by 50; by 75. 

25 is | of 100. Multiplying by 25 gives a product \ as great 
as the product obtained by multiplying by 100 . 

Hence, to multiply by 25, annex two zeros to the multi¬ 
plicand (that is, multiply by 100) and divide by 4. 

50 is 5 of 100. 

Hence, to multiply by 50, annex two zeros to the multi¬ 
plicand and divide by 2 . 

75 is f of 100. 

Hence, to multiply by 75, annex two zeros to the multipli¬ 
cand, divide by 4, and multiply the result by 3. 

2 . Multiply 475 by 125; by 250. 

8)475000 125 is f of 1000. To multiply by 1000, annex three zeros. 

The product is now 8 times as large as it should be. Therefore, 
oyo/o by g obtain the correct result. 

4)475000 j n manner> 250 is | of 1000. To multiply by 250, annex 
118750 three zeros to the multiplicand and divide by 4. 


4)41300 

10325 

2 )41300 

20650 

4)41300 

10325 

3 

30975 


106. Find the products mentally where possible: 


1. 24X25 

2. 64X25 

3. 88X25 

4. 94X50 

5. 53X50 


6. 87X50 

7. 99X75 

8. 103X25 

9. 137X50 

10. 246X75 


11. 824X125 

12. 938X250 

13. 728X500 

14. 1498X125 

15. 2387X25 


16. 7296X250 

17. 8974X750 

18. 4976 X.75 

19. 8397X250 

20. 17986X125 


107. To find the product of two factors when each ends in 5. 
l. Multiply 45 by 45. 45X45 = 2025. 

To find the square of numbers ending in 5, write 25 for the right-hand two 
figures of the product. Next add 1 to the tens’ figure (4 + 1 = 5 in this 
example), multiply by the other tens’ figure (4X5 = 20), and write the 
product next to the 25 already written. The result is 2025. 








MULTIPLICATION 


59 


2. Multiply 35 by 55. 35 X 55 = 1925. 

If the sum of the tens’ figures is even, write 25 as the first part of the product. 
The sum of the tens’ figures (5+3 = 8 ) is even. Take \ of the sum of the 
tens’ figures (£ of 8 = 4) and add it to the product of the tens’ figures multi¬ 
plied together. 5 X 3 = 15; 15 + 4 = 19. Write 19 in the product, making 
1925. 

3. Multiply 45 by 115. 45X115 = 5175. 

The sum of the tens here (4 + 11 = 15) is odd; hence write 75 as the first 
part of the product. Take f the sum of the tens’ figures (£ of 15 = 7£), dis¬ 
regard the fraction, and add the half sum to the product of the tens’ figures 
multiplied together. (4 X 11) + 7 = 51. Write 51 in the product, making 

5175. 

108. Practice on the following until you can read the products 
at sight; 


1. 15X15 

8. 85X85 

15. 95X85 

22. 125 X 45 

2. 25X25 

9. 95X95 

16. 65X55 

23. 125X125 

3. 35X35 

10. 85X65 

17. 45X35 

24. 105 X 35 

4. 45X45 

li. 85X55 

18. 75X85 

25. 95X105 

5. 55X55 

12. 45X65 

19. 95X15 

26. 105X105 

6. 65X65 

13. 55X75 

20. 85 X 45 

27. 115X115 

7. 75X75 

14. 85X35 

21. 65X75 

28. 115X 55 

109. To square any number of two figures. 



1. Multiply 53 by 53. 53X53 = 2809. 

53 is 3 more than 50; 53 plus 3 is 56. Hence, 53X53 = (50X56) 
4 -(3X3) =2809. _ 

[(53)' = (50 + 3) 1 2 = (50 X 50) + (2 X 3 X 50) +(3X3) = (50 X 50) + 
(6 X 50) + (3 X 3) = (50 X 56) +9 = 2809.] 

2. Multiply 69 by 69. 69X69 = (70X68) + 1 2 = 4761. 

69 is 1 less than 70. 69 minus 1 = 68. Hence, 69 X69 = (70X68) 
+ (1X1) =4761. _ 

[(69) 2 = (70 - l) 2 = (70 X 70) - (2 X 1 X 70) + (1X1) = (70X70) - 
(2 X 70) + (1X1) = (68 X 70) +1 = 4761.] 

110. Write the squares of all the numbers from 40 to 60. 




60 


MULTIPLICATION 


111. To find the product of any two numbers of two figures 
each when the units or the tens are alike. 

1. Multiply 42 by 72. 42 X 72 = 3024. 

2X2=4; write 4. Take the sum of the tens’ figures, and multiply by one 
of the units’ figures. (7 + 4) X 2 =22; write 2, and carry 2. Take the prod¬ 
uct of the tens, and add 2. 7 X 4 + 2 = 30; write 30, making 3024. 

2. Multiply 37 by 34. 37X34 = 1258. • 

4X7 = 28; write 8, carry 2. Take the sum of the units’ figures, and mul¬ 
tiply by one of the tens’ figures. (7 +4) X 3 = 33; 33 + 2 = 35; write 5, 
carry 3. Take the product of the tens’ figures, and carry the 3; 3 X 3 + 3 = 12. 
Therefore, the product is 1258. 

Note. The student should take careful notice of the fact that when the 
tens are alike, the units’ figures are added, and vice versa. 


112 . Multiply: 


1 . 

43 

by 

33 

7. 

74 

by 

84 

13. 

93 

by 

98 

19. 

67 

by 

63 

2. 

24 

by 

44 

8. 

56 

by 

86 

14. 

69 

by 

63 

20. 

72 

by 

92 

3. 

52 

by 

62 

9. 

37 

by 

32 

15. 

67 

by 

69 

21. 

87 

by 

46 

4. 

67 

by 

27 

10. 

48 

by 

45 

16. 

78 

by 

76 

22. 

55 

by 

57 

5. 

83 

by 

53 

11. 

58 

by 

54 

17. 

39 

by 

49 

23. 

39 

by 

38 

6. 

69 

by 

29 

12. 

88 

by 

87 

18. 

58 

by 

57 

24. 

46 

by 

36 


113 . To multiply by 11 or by a multiple of 11. 
l. Multiply 287 by 11. 


287X11=3157. 

287 

11 

287 

287 

3157 


By referring to the solution in the margin, it is seen 
that the first figure in the product is 7, the units’ 
figure of the multiplicand; the second figure of the 
product is the sum of the units and the tens; the third 
is the sum of the tens and the hundreds, including the 
carrying figure, and the fourth is the hundreds’ figure 
of the multiplicand plus the carrying figure. 


The principle involved in the preceding example applies to prob¬ 
lems in which it is desired to multiply by a multiple of 11. 




MULTIPLICATION 


61 


2. Multiply 329 by 44. 329 X44 = 14476. 

44 is 4 times 11. Make the multiplication mentally by 11, and multiply the 
result by 4 at the same time. In multiplying by 11 the first figure of the 
product would be 9, which multiplied by 4 gives 36. Write 6, and carry 3. 
Continue the multiplication byll:9-f-2 = ll;4Xll =44, which with 

3 to carry, makes 47; write 7, carry 4. 2 -f 3 = 6; 4 X 5 = 20, which, with 

4 to carry, makes 24; write 4, carry 2. 4 X 3 + 2 = 14; write 14, making 

14476. 


114 . Multiply, mentally, each of the following numbers by 11: 


1. 

54 

5. 96 

9. 244 

13. 396 

17. 4872 

21. 4507 

2. 

35 

6. 58 

10. 372 

14. 927 

18. 3976 

22. 5240 

3. 

43 

7. 39 

ll. 628 

15. 489 

19. 8498 

23. 3007 

4. 

78 

8. 76 

12. 579 

16. 767 

20. 2976 

24. 9080 


115 . Multiply each of the above numbers by 22; 33; 55; 77; 88. 


116 . To multiply by 21, 31, etc., by 101, 201, etc., and by 129, 
561, etc. 

1. Multiply 287 by 41. 

In multiplying by a number having a figure 1 in the 
2o7 41 unit’s place, write the multiplier to one side of the mul- 

1148 tiplicand instead of under it. This plan saves re-writing 

Hygy the figures of the multiplicand as part of the product 

because to multiply by 1 is merely to repeat the figures 
which have already been written. Multiply by the ten’s figure, 4, and write 
the result as indicated. 

2. Multiply 458 by 601. 


458 

2748 

275258 


As in the preceding exercise, write the multiplier to one 
side rather than under the multiplicand. Multiply by 
the hundred’s figure, 6, and write the result as shown. 


3. Multiply 537 by 157. 


537 

2685 

3759 

84309 


157 Write the multiplier to one side as before. Multiply 
first by 5, the ten’s figure, writing the result one place to 
right as indicated. Next multiply by the unit’s figure, 
7, and write the result one place further to the right, as 
illustrated. 





62 


MULTIPLICATION 


Multiply, using the plan illustrated on the preceding page: 


1. 

234 

by 

21 

11. 746 by 701 

21. 834 by 341 

31. 746 by 

146 

2. 

456 

by 

41 

12. 486 by 501 

22. 857 by 451 

32. 735 by 

178 

3. 

728 

by 

81 

13. 839 by 401 

23. 968 by 561 

33. 647 by 

149 

4. 

342 

by 

71 

14. 395 by 801 

24. 576 by 431 

34. 482 by 

163 

5. 

376 

by 

61 

15. 692 by 901 

25. 832 by 741 

35. 965 by 

158 

6. 

748 

by 

31 

16. 748 by 301 

26. 935 by 371 

36. 869 by 

143 

7. 

839 

by 

91 

17. 689 by 601 

27. 347 by 521 

37. 456 by 

145 

8. 

747 

by 

51 

18. 937 by 701 

28. 747 by 671 

38. 846 by 

193 

9. 

856 

by 

61 

19. 437 by 6001 

29. 873 by 771 

39. 695 by 

148 

10. 

645 

by 

81 

20. 748 by 8001 

30. 748 by 671 

40. 853 by 

175 

117. 

To 

multiply by numbers 

i from 11 to 19; 1 

91 to 99; 101 to 


109, etc 

Multiply 583 by 13; by 17. 
5830* 13 
1749 
7579 


583* 

4081 

9911 


17 


Write the multiplier to the right. Multiply by the unit's figure 
and place the result as shown. 

Multiply 719 by 93; by 99. 

(100X719) =71900* 93 (100X719) =719* 99 

(7X719) = 5033 (1X719) = 719 

66867 71181 

First, multiply by 100. (Annex two zeros). Then deduct 
7X719, because 93 is 7 less than 100 
In the second case deduct 1X719, etc. 

Multiply 847 by 103; by 107. 

84700* 103 847* 107 

2541 


847* 

5929 

90629 


87241 

Write the multiplier to one side and multiply only by the unit's 
figure, placing the result as illustrated. 

*The writing in of the zeros in these places is optional. The result is 
the same if the other figures are written in the correct position. 






MULTIPLICATION 


63 


Multiply, using the plan illustrated on the preceding page: 


1. 543 by 12 

li. 867 by 94 

21. 457 by 105 

2. 467 by 17 

12. 389 by 98 

22. 987 by 102 

3. 230 by 14 

13. 430 by 93 

23. 745 by 106 

4. 876 by 17 

14. 245 by 95 

24. 923 by 101 

5. 325 by 13 

15. 442 by 91 

25. 654 by 103 

6. 634 by 16 

16. 331 by 92 

26. 432 by 102 

7. 289 by 17 

17. 346 by 99. 

27. 231 by 104 

8. 980 by 15 

18. 239 by 98 

28. 346 by 107 

9. 847 by 16 

19. 456 by 91 

29. 289 by 109 

10. 455 by 13 

20. 782 by 93 

30 . 319 by 101 


118. To multiply by a number composed of factors. 
1. Multiply 369 by 497. 


First multiply by 7 in the usual -way. The other part of the 
multiplier, 49, is 7 times 7; hence instead of multiplying 369 by 
49, multiply 2583 by 7, placing the result as shown in the illustra¬ 
tion, since the multiplier is 49 tens. Then add the partial 
products. 

2 . Multiply 657 by 321. 

657 

321 First multiply by 3, placing the result as shown. 21 i3 7 times 
' 3. Multiply the result already found by 7, writing the product 

two places to the right of the first partial product, since the 3 rep- 
13797 resents hundreds and the 21, units . 

210897 

119. To multiply any two numbers of two figures. 

Multiply 46 by 92. 46X92=4232. 

First, 2 X 6 = 12; write 2, carry 1. Next, 2X4 + 1 (carried) = 9; also 
9 X 6 = 54. Add 54 and 9, giving 63; write 3, carry 6. Next, 9X4 + 
(carried) = 42, which makes a completed product of 4232. 

CHECKING THE WORK 

120. The following checks will be helpful in proving multi¬ 
plication. 


369 

497 

2583 

18081 

183393 





64 


MULTIPLICATION 


121 . 1 . Divide the product by one of the factors: the quotient 
will be the other factor. If there are more than two factors, the 
product divided by one of them will give a quotient equal to the 
product of all the other factors. 

(a) 18X25 = 450. Proof. 450-^18 = 25. 

(5) 4X8X9X5 = 1440. Proof. 1440^4 = 360 = 8X9X5. 

122. Casting out the nines. 


The excess of nines in the multiplicand is 5, and in the 
multiplier is 4. The product of the excesses 4 and 5 is 20, 
and the excess of nines in 20 is 2. The excess of nines in 
the product, 902, is 2. Hence the multiplication is prob¬ 
ably correct. 

The excess of nines in the product is equal to the excess of nines 
in the product of the excesses. 

123. Casting out the elevens. 

The excess of elevens in 248 is 6, (8—4 + 2); the 
excess of elevens in 37 is 4, (7 — 3). The product of the 
excesses is 4 X 6, or 24, and the excess of elevens in 24 is 
2. The excess of elevens in 9176 is 2, (6 + 1 + 11 — 7 
—9). Since the results agree, the multiplication is 
probably correct. 


124. 

Multiply, and check the results. 

Use the 

check 

assigned 

by the teacher. 









l. 

43 

by 29 

10. 

875 

by 

756 

19. 

64 

by 

36 

2. 

72 

by 37 

n. 

1249 

by 

648 

20. 

93 

by 

67 

3. 

48 

by 27 

12. 

3147 

by 

639 

21. 

94 

by 

69 

4. 

37 

by 84 

13. 

2489 

by 

763 

22. 

81 

by 

74 

5. 

43 

by 56 

14. 

3271 

by 

654 

23. 

82 

by 

29 

6. 

91 

by 28 

15. 

5298 

by 

324 

24. 

34 

by 

85 

7. 

96 

by 47 

16. 

6476 

by 

549 

25. 

35 

by 

63 

8. 

56 

by 46 

17. 

8749 

by 

624 

26. 

73 

by 

57 

9. 

83 

by 34 

18. 

9483 

by 

424 

27. 

79 

by 

67 


248= 6 
37= 4 
1736 24 = 2 
744 

9176 = 2 


41 = 5 
22 = 4 _ 

82 20 = 2 
82 

902 = 2 


DIVISION 


125. Division is the process of finding how many times one num¬ 
ber is contained in another. 

SHORT METHODS 

126. To divide by 10, 100, 1000, etc. 

Divide 4370 by 10. 

The solution at the left shows that to divide by 10 
4o /U — 1U — 4o7 ^ j s necessary only to move the decimal point one 

437.0 place to the left. 

10)4370 I n manner, to divide by 100, 1000, etc., move 

the point 2 places, 3 places, etc., to the left. 

127. Divide each of the following numbers by 10, 100, 1000, 

10,000 


l. 130000 

6. 24750000 

ll. 3489 

2. 1250000 

7. 400000 

12. 7248 

3. 40000 

S. 1800000 

13. 5673 

4. 750000 

9. 2000000 

14. 8470 

5. 1160000 

10. 14500000 

15. 19872 


128. To divide by 25, 50, 75, etc. 

In many cases the work of division can be lessened by making 
the operation one of multiplication. 

Divide 2800 by 25. 28004-25 = 28X4 = 112. 

25 is 1 of 100. Divide 2800 by 100 by dropping the two zeros. But, in 
dividing by 100, you have divided by a number 4 times too large, consequently 
the quotient is only 1 as large as it should be (Principle 17, page 117). To 
correct the error, multiply the 28 by 4. 

Note. Further application of the principle illustrated in the last example 
is made in the chapter on Aliquot Parts, page 29. 

VAN TUYL’S NEW COMP. AR —5 65 



6G 


DIVISION 


129. Divide: 

1. 1400 by 25 

2. 1600 by 25 

3. 1800 by 50 

4. 2400 by 50 


5. 142 by 25 

6. 153000 by 250 

7. 72000 by 500 

8. 16000 by 500 


9. 27000 by 75 

10. 36000 by 250 

11. 48000 by 500 

12. 54000 by 750 


130. To divide by a number composed of factors, 

l. Divide 12726 by 63. 

9X7 = 63. 

9)12726 

7) 1414 The factors of 63 are 9 and 7. Instead of making a long 

L _ division, divide by the factors, 9 and 7, using short division. 

202 


2 . Divide 38971 by 64. 


8X8 = 64. 
8)38971 

8) 4871 3 rem. 

608 7 rem. 
7X8+3 59 rem. 


The factors of the divisor are 8 and 8. Divide 
by the factors as before, reserving the remainders 
as shown. To find the remainder, multiply the 
first divisor, 8, by the second remainder, 7, and 
add the first remainder, 3. Thus, 7X8+3 = 
59, remainder. Hence the result is 608, with a 
remainder of 59. 


3. Divide 174684 by 504. 
7X8X9 = 504. 


7 ) 174684 

8) 24954 6 rem. 

9) 3119 2 rem. 

346 5 rem. 

5X8X7 = 280 
2X7= 14 

6 =_ 6 

300 

300, remainder. 


The factors of 504 are 7, 8, and 9. Divide by 
the factors as before. The quotient is 346. To 
find the remainder, multiply the third remainder 
by the first and second divisors, 7, and 8, (5 X 8 
X 7 = 280). Next multiply the second re¬ 
mainder by the first divisor, 7, (2X7 = 14). 
To the sum of these two products, add the first 
remainder, 6, giving 300 as the remainder: 









CHECKING THE WORK 


67 


4. Divide 58750 by 2400. 

The factors are 4, 6, and 100. Divide by 100 
by “cutting off” the right-hand two figures of 
the dividend. Then divide by the factors 4 and 
6 . The quotient is 24. To find the remainder 
first treat the divisors and remainders as though 
the divisor, 100, and the remainder, 50, were not 
present. 2X4+3 = 11. To complete the 
remainder, annex the remainder, 50, to the 11, 
giving a remainder of 1150. 

CHECKING THE WORK 

131. The following methods will be found helpful in checking 
the work. 

1. Multiply the divisor by the quotient and add the remainder 
(if any); the result should equal the dividend. 

2. Cast out the nines. 

8879 4- 247 = 35, with a remainder of 234. 

Therefore, 8879 = 247X35+234 
The excess of 9’s in 8879 is 5. 

The excess of 9’s in 247 is 4. 

The excess of 9’s in 35 is 8. 

The excess of 9’s in 234 is 0. 

4x8+0 = 32. 

The excess of 9’s in 32 is 5. 

Since the excess of 9’s in the dividend is equal to the excess of 9’s in the 
product of the excesses in the divisor and quotient plus the excess in the 
remainder, the division is probably correct. 

3. Cast out the elevens. 

3504-i-54 = 64, with a remainder of 48. 

Therefore, 3504 = 54X64+48 
The excess of ITs in 3504 is 6. 

The excess of ITs in 54 is 10. 

The excess of ITs in 64 is 9. 

The excess of ITs in 48 is 4. 

10X9+4 = 94. 

The excess of ITs in 94 is 6. 

Since the excesses agree, the division is probably correct. 


4X6X100 = 2400. 
4 )587.50 
6)146 3 rem. 

24 2 rem. 
2X4+3=11, 
1150, remainder. 




68 


DIVISION 


132. Divide and check the work. Use the check assigned by 
the teacher. 


1 . 

4872 by 36 

8. 24763 by 108 

15. 247689 by 4900 

2. 

7349 by 42 

9. 45972 by 54 

16. 324798 by 5600 

3. 

5987 by 56 

10. 79489 by 88 

17. 498749 by 5400 

4. 

7879 by 72 

11. 48900 by 3500 

is. 897200 by 3600 

5. 

9478 by 84 

12. 147260 by 3600 

19. 948720 by 48000 

6. 

9764 by 96 

13. 248750 by 4200 

20 . 672000 by 24000 

7. 

11492 by 99 

14. 329764 by 4800 

21 . 486000 by 18000 


DIVISION BY CONTINUED SUBTRACTION 

133. This method of division is often required in civil service 
examinations. 


134. By the method of continued subtraction divide 784 by 
149, and prove the work. 


784 

l. 149 


635 


2. 149 
486 

3. 149 
337 

4. 149 


Subtract the divisor from the dividend, and from the succes¬ 
sive remainders, until a remainder less than the divisor is 
obtained. The number of subtractions is the quotient. 

Proof. 

5 X 149 -f- 39 = 784; or, 784 -r- 149 = 5, with a remainder 
of 39. 


188 
5. 149 


39 


135. Divide by continued subtraction and prove: 


1. 489764 by 42875 

2. 447859 by 48727 

3. 128976 by 54738 

4. 249768 by 55987 

5. 473964 by 75428 

6. 963874 by 85925 


7. 389726 by 98762 

8. 598370 by 87820 

9. 397645 by 98435 

10. 947680 by 94768 

11. 247682 by 31463 

12. 437263 by 48721 


FACTORING 


136. 6 is the product of what numbers? 

Because 2 X 3 = 6, 2 and 3 are called the factors of 6. 

137. What are the factors of 10? 15? 18? 21? 

138. A prime number is one not exactly divisible by any num¬ 
ber except itself and 1. 

139. Prime factors are prime numbers. 

140. Factoring is the process of separating a number or quantity 
into its factors. 

Remark. Factoring is important, not merely for the purpose of resolving 
a given number into its factors, but also for its assistance in the solution 
of problems. In many problems in fractions, practical measurements, and 
percentage, in fact, in all problems in which cancellation is used, the work 
involved in the solution is very materially lessened by the ability to see fac¬ 
tors quickly. 

The ability quickly to factor a given number depends upon a knowledge 
of the tests of divisibility, with which the student should become thoroughly 
familiar. 

141. Tests of Divisibility. 

1. All even numbers are divisible by 2. 

Thus, 4, 8, 18, 38, 96, 144, are divisible by 2. 

2. If the sum of the digits of any number is divisible by 3, the 
number is divisible by 3. 

Thus 144957 is divisible by 3 because 1+4 + 4+ 9+ 54*7 = 30, and 
30 is divisible by 3. 

3. All numbers whose right-hand two figures are zeros, or 
express a number divisible by 4, are divisible by 4. 

Thus, 100, 144, and 3288, are divisible by 4. 

4. All numbers whose units 7 figure is either a zero or 5 are 
divisible by 5. 

Thus, 60 and 135 are divisible by 5. 

69 


70 


FACTORING 


5. All even numbers the sum of whose digits is divisible by 3, 
are divisible by 6. 

Thus, 126, 918, and 45252 are divisible by 6. 

In practice, notice first whether the number is even. If so, 
proceed as in test for 3. 

6. 7, 11, and 13 will divide 1001 and any of its multiples; as, 
3003, 8008, 12012, etc. 

7. All numbers whose right-hand three digits are zeros, or 
express a number divisible by 8, are divisible by 8. 

Thus, 3000, 6624, and 18232 are divisible by 8. 

8. All numbers, the sum of whose digits is divisible by 9, are 
divisible by 9. 

Thus, 45621 and 234819 are divisible by 9. 

9. All numbers whose right-hand figure is 0 are divisible by 10. 

Thus, 60, 140, 1190, are divisible by 10. 

10. All numbers- whose right-hand two figures are zeros, or 
express a number divisible by 25, are divisible by 25. 

Thus, 9800, 4125, and 875 are divisible by 25. 

11. All numbers whose right-hand three figures are zeros, or 
express a number divisible by 125, are divisible by 125. 

Thus, 3000, 4625, and 3375 are divisible by 125. 

142. Applying the tests given above, find the prime factors of 
12,464. 

8)12464 
2 ) 1558 

1 9) 779 Therefore, 2 X 2 X 2 X 2X19 X41 = 12,464. 

41 

The “test” for 8 ehows that 8 is a factor. The quotient 1558 is even whence 
2 is a factor. By trial 19 is found to be a factor, resulting in a quotient of 41, 
a prime number. 

8 was used as a divisor first to save time. Its prime factors are known to be 
2 X2X2. Therefore, the prime factors of 12,464 are 2X2X2X2X19 
X 41. 

143. There is no regular method of determining large factors. 
Unless one has access to factor tables, nothing remains but to try 
the prime numbers until the right one is found. There are, how¬ 
ever, two fundamental principles of factoring that should be 
borne in mind: 





GREATEST COMMON DIVISOR 


71 


1. A factor of a number is also a factor of all the multiples of 
that number. 

Thus, 7 is a factor of 21. 7 is also a factor of 42, 63,105,189, 420,1071, all 
of which are multiples of 21. 

2. A common factor of two numbers is a factor also of the sum 
or of the difference of those two numbers. 

Thus 4 is a common factor of 20 and 36. It is a factor also of 56 (20 + 36), 
and of 16 (36 - 20). 


144 . Find the prime factors of the following numbers: 


1. 

48 

7. 729 

13. 2664 

19. 

16296 

25. 

5872 

2. 

64 

8. 343 

14. 4064 

20 . 25728 

26. 

9536 

3. 

80 

9. 1125 

15. 4512 . 

21 . 

6875 

27. 

3856 

4. 

105 

io. 3375 

16. 3824 

22 . 

5375 

28. 

7328 

5. 

144 

ll. 3654 

17. 9664 

23. 

4004 

29. 

15309 

6. 

160 

12 . 3248 

18. 9750 

24. 

3224 

30. 27783 


GREATEST COMMON DIVISOR 

145 . The greatest common divisor of two or more numbers 
is the largest number that will exactly divide each of them. 

146 . The only practical application (so far as this work is con¬ 
cerned) of the principles involved in finding the G. C. D. is in 
reducing common fractions to their lowest terms. 

GREATEST COMMON DIVISOR BY INSPECTION 

147 . In many cases the G. C. D. of two or more numbers can 
be determined by inspection by observing the following: 

Principles. 1. The G. C. D. of two or more numbers cannot be 
greater than the smallest number in the list. 

2. The G. C. D. of two or more numbers cannot be greater than 
the smallest difference between any two numbers in the list. 

3. The G. C. D. of two or more numbers is either (a) the smallest 
number, (b ) the smallest difference, of (c) a factor of one of them. 


72 


LEAST COMMON MULTIPLE 


148 . l. Find the G. C. D. of 24, 72, 120, and 168. 

By inspection it is seen that the smallest number, 24, will divide each of 
the several numbers, hence it is the G. C. D. 

2. Find the G. C. D. of 70, 168, 224, 252. 

The smallest number is 70. The smallest difference is 28 (252 — 224). 
By inspection 28 is seen not to be»a divisor of 70. 28 = 2 X 14. Since 28 is 
not a common divisor, either 2 or 14 must be eliminated. Since the greatest 
common divisor is sought, eliminate the 2. 14 is seen to be a divisor of each of 
the numbers. Hence 14 is the G. C. D. 


149. If the G. C. D. cannot be determined by inspection 
proceed as follows: 

l. Find the G. C. D. of 188, 470, 705, and 940. 


188-470-705-940 
188- 94-141- 0 
0- 94- 47- 0 
47 is the G. C. D. 


Divide all the other numbers by the smallest 
number, writing the remainders as shown in 
the solution. The “divisor” must also be 
“brought down.” Divide again by the small¬ 
est number. By inspection, it is now seen that 
47 is the G.C.D. 


150. Find, by inspection when possible, the G. C. D. of the 
following numbers. 


1. 912 and 1121 

2. 296 and 407 

3 . 792 and 1016 

4 . 629 and 1147 

5 . 387 and 559 

6. 319 and 551 

7 . 85 and 119 

8. 145 and 206 

9 . 2813 and 3589 


10. 52, 91, and 143 

11. 961, 1271, and 372 

12. 68, 102, and 187 

13 . 2167, 2561, and 985 

14 . 3749, 8313, and 1141 

15 . 48, 72, 120, and 288 

16 . 1681, 2501, and 410 

17 . 6859, 11,191, and 3610 

18 . 90, 108, 144, and 111 


LEAST COMMON MULTIPLE 

151. A multiple of any given number is a number that will 
contain the given number without a remainder. Thus, 24 is a 
multiple of 4. 





LEAST COMMON MULTIPLE 


73 


152. 24 is a common multiple of 4 and 6, because it bontains 4 
and 6 each without a remainder. 


153. 24 is the least common multiple of 4, 6, 8, and 12, because 
it is the smallest number that will contain them without re¬ 
mainders. 


LEAST COMMON MULTIPLE BY INSPECTION 

154. In the great majority of practical cases the L. C. M. can 
be found by inspection by observing the following: 

Principles., l. The L. C. M. of two or more numbers cannot be 
smaller than the largest number in the list. 

2 . The L. C. M. of two or more numbers is either (a) the largest 
number , or (6) a multiple of the largest number. 

Find the L. C. M. of 9, 16, and 24. 

24, the largest number, is not the L. C. M. Hence, try 48, 72, 96, etc., 
until a number is found that is a common multiple. By trial it is found to 
be 144. 

If the L. C. M. is not readily discovered by inspection, pro¬ 
ceed as follows: 

Find the L. C. M. of 6, 8, 12, 15, 20, 24, 30, and 36. 

36 | 6-8-1^-10-20-24-30-36. 

3-2-5-1. 


36X2X5 = 360. 

Write all the numbers in one line. Cancel all the numbers that are con¬ 
tained in any of the other numbers. Divide the remaining* numbers by the 
largest number in the list having a factor common to one or more of the other 
numbers. Use 36. The G. C. D. of 36 and 20 is 4. 20 divided by 4 is 5. 
Write 5. The G. C. D. of 36 and 24 is 12. 24 divided by 12 is 2. Write 2. 
And so on through the list of numbers. A new line of numbers is now ob¬ 
tained. Proceed as at first. Cancel all numbers contained in some other 
number in the line. Divide as before if necessary, and continue the process 
until the remaining numbers have no common factor except unity. The 
L. C. M. is the product of all the divisors and the remaining numbers. 





74 


CANCELLATION 


155. Find by inspection, when possible, the L. C. M. of: 


1. 2, 4, 6, 12, 15. 

2. 3, 6, 9, 18, 30. 

3. 5, 12, 15, 30, 40. 

4. 8, 12, 24, 60, 84. 

5. 15, 25, 36, 75, 100. 


6. 13, 15, 25, 65, 75. 

7. 24, 36, 48, 42, 120. 

8. 6, 20, 48, 100, 144. 

9. 9, 16, 64, 80, 128. 

10. 36, 45, 72, 105, 150. 


CANCELLATION 


156. Problems frequently arise in which the solution is best 
made like the following: 


Divide the product of 36X18X21 by the product of 9X28X12. 


Write the factors of the dividend above a 
horizontal line, and the factors of the divisor 
below the line, as shown in the solution. 9, 
below the line, is contained in 18, above the 
line. Both numbers are canceled, and the 
quotient, 2, is written near the 18. In the 
same way 12 is contained in 36, 3 times. Since 28 is not divisible by 21, both 
numbers are divided by their common factor, 7. w The numbers 21 and 28 are 
both canceled, and the quotients, 3 and 4, are written near the numbers, re¬ 
spectively. 2 of the dividend now divides the 4 in the divisor; that is, the 
common factor, 2, is canceled in both dividend and divisor. (See Principles of 
Division, No. 21, page 117.) The product of the remaining factors above the 
line is now divided by the product of the factors left below the line, thus, 


3 2 3 

30X18X21 _ 9 
9X28X12 2 
4 
2 


3X3 _ 9 
2 2 


4}, the desired quotient. 


Using cancellation, divide: 

1. 8X6X12X15 by 3X4X5X18. 

2. 15X12X24X39 by 10X13X4X16. 

3. 125X80X96X216 by 50X60X32X540. 

4. 48X56X63X72 by 36X78X42X108. 

5. 30X45X9X12 by 5X50X18X60. 

6. 6X12X80X20 by 4X9X75X120. 

7. 8X24X36X160 by 12X18X30X60. 

8. 36X45X54X75X68 by 144X15X36X51X65. 

9. 57X39X76X91X96 by 114X65X95X160. 

10. 38X64X68X69X217 by 133X12X17X460. 



CANCELLATION 


75 


11. How many tubs of butter averaging 30 lb. each, at 56^ 
per pound, will pay for 8 pieces of muslin, each piece containing 
49 yd. at 30 ^ per yard? 


The cost of the muslin is represented by the product of 8 X 49 X $.30; 
the value of 1 tub of butter, by the product of 30 X $.56. The number of 
tubs of buttef required is found by dividing the cost of the muslin by the 
value of 1 tub of butter. Thus: 


7 

8X49X-30 

30X.50 

7 


number of tubs of butter. 


12. How many pounds of sugar at 15 £ a pound can be had in 
exchange for 20 doz. eggs at 48 £ per dozen? 

13. At $1.20 a bushel, how many bushels of potatoes will pay 
for 3 chests of tea, weighing 36 lb. each, at 40 ^ a pound? 

14. How many boys earning 20 £ an hour, 8 hours a day, in 15 
days, will earn as much as 4 men at 60 an hour, 10 hours a day, 
in 18 days? 

15. If $560 earns $84 interest in 2 yr. 6 mo., how much interest 
will $640 earn in 3 yr. 6 mo., at the same rate? 

16. At $3.60 per barrel, how many loads of apples, 21 bbl. to 
the load, will be required to pay for 18 bbl. of sugar, averaging 
350 lb. to the barrel, at 6 ^ per pound? 

17. If 17 tons of coal cost $178.50, how much will 51 tons cost 
at the same rate? 

18. Find the cost of 40 tubs of butter, each weighing 56 lb., 
if 28 tubs of the same quality, weighing 48 lb. each, cost $698.88. 

19. How many yards of cloth worth 70 i a yard, would be given 
in exchange for 4 loads of oats, each load containing 40 bags of 
2 bu. each, if oats are worth 84 i a bushel? 

20. How many cases of eggs, each containing 30 doz., at 42^ 
a dozen, will pay for 6 bbl. of sugar averaging 350 lb. to the bar¬ 
rel, at 6 £ a pound? 



POSTAL RATES 

157 , First-Class Matter includes letters, postal cards, post¬ 
cards, and all matter wholly or partly in writing, whether sealed 
or unsealed (except manuscript copy) and all other matter sealed 
or closed against inspection. 

The limit of weight of first class matter is 70 pounds. 

Rates of Postage. Postal cards and post cards 1 cent each. 
All other first-class matter 2 cents for each ounce or fraction 
thereof. 

Second-Class Matter includes newspapers and periodicals 
bearing notice of entry as second-class matter. 

No limit of weight is prescribed. 

Rate of Postage. One cent for each 4 ounces or fraction thereof. 

Third-Class Matter includes circulars, newspapers and peri¬ 
odicals not admitted to second-class, manuscript copy, proof 
sheets, photographs, and other miscellaneous printed matter. 

The limit of weight is 4 pounds. 

Rate of Postage. One cent for each 2 ounces or fraction thereof. 

Fourth-Class Matter, or Domestic Parcel Post mail includes 
merchandise, farm and factory products, seeds, plants, etc., 
books and catalogues, miscellaneous printed matter weighing 
more than 4 pounds, and all other mailable matter not embraced 
in the first, second, or third classes. 

Parcel post matter must not be of a kind likely to injure the 
person of any postal employee or to damage the mail equipment 
or other mail matter, nor must it be of a character perishable 
within a period reasonably required for transportation and 
delivery. 

Rates of postage on domestic parcel post matter are as follows: 

(a) Parcels weighing 4 oz. or less, except books, seeds, plants, 
etc., 1 i for each ounce or fraction thereof, for any distance. 

76 


POSTAL RATES 


77 


(6) Parcels weighing 8 oz. or less containing books, seeds, plants, 
etc., 1 £ for each 2 oz. or fraction thereof, for any distance. 

(c) Parcels weighing more than 8 oz. containing books, seeds, 
plants, etc., parcels of miscellaneous printed matter weighing more 
than 4 lb., and all other parcels of domestic parcel post, or fourth 
class matter weighing more than 4 oz. are chargeable according 
to distance or zone, at the pound rates, a fraction of a pound being 
considered a full pound. 

Rates of Postage in the Different Zones 

Local rate, 1st pound, 5 each additional 2 pounds, 1 j£. 

1st and 2d zones, 1 to 150 miles, 1st pound, 5 each additional pound, 1 jl. 

3d zone, 150 to 300 miles, 1st pound, 6 each additional pound, 2 

4th zone, 300 to 600 miles, 1st pound, 7 each additional pound, 4 jf. 

5th zone, 600 to 1000 miles, 1st pound, 8 j£; each additional pound, 6 jf. 

6th zone, 1000 to 1400 miles, 1st pound, 9 j£; each additional pound, 8 

7th zone, 1400 to 1800 miles, 1st pound, 11 each additional pound, 10 

8th zone, over 1800 miles, 1st pound, 12 jf; each additional pound, 12 

The limit of weight for local delivery and for the first three 
zones is 70 pounds; for all other zones, 50 pounds. Parcel post 
matter may not exceed 84 inches in length and girth combined. 

Insurance may be secured by a payment of 3 £ extra for an 
amount up to $5; 5 £ up to $25; 10 up to $50; 25 i up to $100. 

Parcels valued at not more than $100 may be sent “C. O. D.” 
upon the payment of a fee of 10 cents, if the amount to be remitted 
does not exceed $50; or a fee of 25 i for greater amounts up to 
$100. This fee includes insurance up to $50 and $100 respec¬ 
tively. The fee for the return, by postal money order, of the 
amount collected is paid by the addressee. 

A parcel post guide and map are used to determine in which 
zone a post office is located from the office of mailing. 

PROBLEMS 

How much postage is required on the following? 

1. A letter weighing l\ ounces. 

2. A sealed package weighing 9f ounces. 

3. A package of newspapers weighing 1 lb, 6 oz. 


78 


POSTAL RATES 


4. A photograph weighing 5J ounces. 

5. A mail order house sends 18 packages each weighing 9 lb. 
4 oz. a distance of 900 miles. 

6. A boy went to a post office to mail 3 letters, one of which 
weighed lj ounces, and the other two \ ounce each, 4 post cards, 
a magazine weighing 8J ounces, and a package of circulars weigh¬ 
ing 7 ounces. 

7. A parcel of merchandise weighing 18 pounds to a post office 
in the fifth zone. 

8. A 40-pound parcel of books to the second zone. 

9. An insured parcel of dry goods weighing 14 pounds 6 ounces, 
valued at $32.75, to a post office in the sixth zone. 

10. A “C. O. D.” insured parcel weighing 12§ pounds, valued at 
$18.50, to a post office in the seventh zone. 

11. A 7-ounce book to a post office in the fifth zone. 

12. A 10-ounce parcel of dry goods to the third zone. 

13. A “C. O. D.” parcel valued at $75 and weighing 20 lb. is 
mailed to the 8th zone. Find the cost of mailing it; also the cost 
to the addressee, including money order fee. (For money order 
rates, see p. 391.) 

14. A man sends a 30-pound pail of butter to a post office in 
the second zone. How much is the postage? 

15. A merchant in Toledo, O., telegraphed a jobber in Chicago, 
“Send C.O.D. by parcel post 100 dozen# 546 buttons.” The but¬ 
tons came billed at 22 cents a dozen. How much did the merchant 
have to pay for the buttons, including the money order fee? 

16. A jeweler in Spokane, Wash., ordered from a wholesaler in 
San Francisco, Cal. (fifth zone), one dozen watches sent C.O.D. 
by parcel post. If the watches were valued at $7.50 each, and the 
parcel weighed 3J pounds, (a) how much was the total cost of 
mailing the watches? ( b ) how much did they cost the jeweler in 
Spokane, including the money order fee? (c) if the watches were 
broken en route, how much insurance could be collected? 


FRACTIONS 

158. What part of a dollar is 50 j£? 25^? 75 124 cf? l(W? 

50? Iff? 

Each of these amounts is less than one dollar. They are frac¬ 
tional parts of a dollar. 

159. A fraction is a part of a unit. 

50 = £ of a dollar = .50 of a dollar. 

25 £ — i of a dollar = .25 of a dollar. 

75 { = l of a dollar = .75 of a dollar. 

5 i = ifo of a dollar = .05 of a dollar. 

M = r4xr of a dollar = .01 of a dollar. 

How many ways are there of writing a fraction? 

Are and .5 equal in value? 

Are f, tVtt, and .75 equal in value? 

Are and .05 equal in value? 

Are and .01 equal in value? 

160. There are, then, two ways of writing a fraction. In the 
common fraction form any number may be used as a denominator 
of the fraction. In the fractions J, f, and 2, 4, 20, and 100, 
respectively, are the denominators. 

161. In the decimal fraction form, the denominator must be 10 
or some power* of 10. In the fractions .5, .75, .05, and .125, the 
denominators are 10, 100, 100, and 1000, respectively. 

162. Into how many parts is this circle 
divided? 

What part of the circle is heavily shaded? 

How much of it is lightly shaded? 

How does the lightly shaded part compare 
in size with the heavily shaded part? I = 2 X i 

*By “power of 10” is meant the product resulting from using 10 a given mumber of times as 
a factor. Thus, 10 X 10, or 100, is the second power of 10; 10 X 10 X 10, or 1000, is the 3d 
power of 10, etc. 



79 




80 


FRACTIONS 


This illustration emphasizes an important principle: 

The larger the denominator , the smaller the fraction. Compare 
with the Principles of Division (No. 17, page 117.) 

What part of the circle is not shaded? 

Which part of the fraction, f, shows the number of parts into 
which the circle is divided? 

Which part shows the number of eighths in the fraction? 

In the fraction, f, 8 is the denominator, and shows that the 
unit (the unit, in this case, is the circle) has been divided into 8 
equal parts. 5 is the numerator, and shows that 5 of the 8 parts 
have been taken to form the fraction, f. 

Important. For all practical purposes, a common fraction repre¬ 
sents a division. 5 -r- 8 = f. These expressions are the same except 
in form. Each may be read “5 divided by 8.” It is clear, there¬ 
fore, that 5, the numerator of the fraction, is a dividend, and 
that 8, the denominator, is a divisor. 

163. Decimal and common fractions are so closely associated in 
business problems, that it has been deemed wise to treat them together. 

There are three things to learn about decimals: (1) The use of 
decimal point. (2) Notation and numeration. (3) The common 
fraction equivalent, and vice versa. 

Into how many equal parts has this cir¬ 
cle been divided? 

One of these equal parts is called one 
tenth, written ^ or .1. 

Into how many equal parts has one of 
the tenths been divided? 

If each of the tenths were divided into 
ten equal parts, into how many smaller 
parts would the entire circle be divided? 

One of these small parts is what part of the circle? ^ of ^ ^ 

or .1 of .1 = .01. (Principles 2 and 4, page 116). 

164. .1 and .01 are called decimal fractions, or, more often- 
decimals. (From the Latin word decern , meaning ten.) They are 
decimals because their denominators are 10 or some power of 10. 

165. The denominator of a decimal is not written, but is indi¬ 
cated by the number of figures used to write the decimal. 





REDUCTION OF FRACTIONS 


81 


166. In reading decimals, the denominator is named the same 
as though it were written. Thus, the decimal .015 is read “fifteen 
thousandths,” the same as if writtenx oos - The denominatoris de¬ 
termined by the fact that the decimal contains 3 figures (“places”). 
For each place, one cipher is written to the right of the figure 1, 
making, in this case, 1000. 

ORAL EXERCISE 

167. Read the following decimals: 


.4 

9. .025 

17. .125484 

.14 

10. .0025 

18. 236.324562 

.05 

11. 15.0125* 

19. .000125 

.125 

12. 28.02345 

20. 10000.0001 

.001 

13. 75.00001 

21. .10101 

.101 

14. .00125 

22. .4755 

.427 

15. .00025 

23. 125.0125 

.821 

16. 112.01875 

24. .0000001 


REDUCTION OF FRACTIONS 

168. How many eighths equal 1? How many quarters equal 
J? How many eighths equal §? 

4 _ 2 _ 1 QTlr l 1 _ 2 _ 4 
8 — 4 — 2) anCl 2—4—¥ 

These equations illustrate what is meant by reduction of frac¬ 
tions. The form of the fraction, f, has been changed so as to read 
“§,” but its value is not changed. 

In the first equation, the fraction, f, has been reduced to its 
lowest terms. 

In the second equation, the fraction, §, has been reduced to 
higher terms. 

169. The terms of a fraction are its numerator and denominator. 

170. A fraction is reduced to its lowest terms when its numer¬ 
ator and denominator cannot both be exactly divided by any 
whole number except 1. 

♦Read “fifteen, and one hundred twenty-five ten-thousandths.” 

VAN TUYL’S NEW COMP. AR.—6 


82 


FRACTIONS 


126 


42 

189“ 

3 

II 

1 CO 

1 CO 


171. l. Reduce fll 1° its lowest terms. 

2 Divide both terms of the traction by any 

— = — common divisor, say 3. The resulting frac- 

21 3 tion i s 42 Divide again by any common 

divisor, as 3, obtaining ff. Now divide both terms by 7, arid the result is §, the 
lowest terms, because the numbers 2 and 3 cannot be divided by the same 
number. 

2. Reduce fff to its lowest terms. 

493 17 this l rac ^ on the common divisor cannot be readily 

29-= — determined by inspection. Find the greatest common 

899 31 divisor by the method explained on page 71. 29 is the 

G. C. D. Dividing both terms of the fraction by 29 gives if, the lowest terms. 

Compare these solutions with the principles of division (No. 21, 
page 117.) 

State the principle involved in the last two examples. 

3. Reduce ff| to its lowest terms. 

688 

From the denominator, 688 , take 2 times the numer¬ 
ator, 301. The remainder, 86, is factored. 86 = 
2 X 43. By inspection, 2 is not a common factor, but 
43 is found, on trial, to be a common factor. Therefore, 
the fraction reduce^, to its lowest terms. 


2X301 = 602 
~86 

86 = 2X43. 
301 _ 7 
688 _ 16 


43 


4. Reduce f to fortieths. 

40 g = 5 The denominator of the fraction desired is 40, which is 

y ^ 5 _ 35 5 times as large as the denominator of the given fraction. 

— — If f is to be changed to a fraction having a denominator 

8 X 5 = 40 5 times as large the numerator also must be 5 times as 

large. Hence, multiplying both numerator and denominator by 5 gives the 
desired results, f &. 

Compare with the principles of division (No. 21, page 117). 

5. Reduce .13 to ten-thousandths. 

10,000^-100=100 require ? deno “i™t°y s 

’ 100 times as large as the de- 

13X100=1300. Therefore, .1300 nominator of the given fraction 
Therefore the numerator should be 100 times as large. Hence, the required 
decimal is .1300. 

Observe that the process of reduction consists in annexing ciphers 

Show the application of the principles of division. 







REDUCTION OF FRACTIONS 


83 


EXERCISE 

172. Reduce the following fractions to their lowest terms (men¬ 
tally, when possible): 

9 355 207 152 

*' 568) '5X2) TTT 


1. 

72 84 96 

■glT) 12 0) "116 

5. 

1240 1640 3088 

1648) 2050)4632 

9. 

2. 

125 48 56 

15 0) 14 4) 16 8 

6. 

1998 5922 2448 

2^9 7) 673 2) 7272 

10. 

3. 

111 156 165 

1 17) 2 0 7) T2^ 

7. 

437 609 341 

5 5 1) 7 8 3) 4 6 5 

11. 

4. 

45 45 55 

(TiT) 18 0) 4 9 5 

8. 

648 244 201 

TT6) 42 7 

12. 


Change to higher terms: 

13. f to 24ths. 

14. | to 48ths. 

15. I to 63ds. 

16. .05 to thousandths. 

17. .04 to hundred-thousandths. 

18. f to 36ths. 

173. To reduce an improper fraction to a whole or a mixed 
number. 

How many halves are there in 1 circle? in 2 circles? in 3 circles? 
How many quarters are there 1 circle? in 2 circles? in 3 circles? 
How many eighths are there in 1 circle? in 2 circles? in 3 circles? 


13. rr to 55ths. 

20 . i, §, J, -§-, to 12ths. 

2R t, I, i tit to 40ths. 

22 . -f-, -g-g-, .5, .8 to lOOths. 

23. f, f, .75, .125 to 24ths. 

24. A, I, *33i .16f to 48ths 





: T "S’* 


— 2 — 4 a* —a 4 »• 3 circles 2 4 "s - * 

ircles = how many circles? f circles = how many circles? 

rc les = how many circles? V circles = how many circles? 

[rcles = how many circles? V 6 circles = how many circles? 

rcles = how many circles? circles = how many circles? 

the fractions in this paragraph are improper fractions, be- 
their numerators are equal to, or greater than, their denom- 
•s. 








84 


FRACTIONS 


174. A fraction having a numerator smaller than its denomin¬ 
ator is a proper fraction. 

175. Improper fractions may be reduced to whole or mixed 
numbers. 

176. A mixed number is the sum of a whole number and a frac¬ 
tion. Thus, 2} is a mixed number; it is the sum of 2 and }. 

177. Reduce Y to a mixed number. 

37-j-5 = 7f. 

There are f in one unit. In V there are as many units as 5 is contained 
times in 37, or 7 times and f over. 

Therefore, V = 7f. 

EXERCISE 

178. Reduce the following improper fractions to whole or 
mixed numbers: 


1. 

4 3 

U 

7. 

8 9 8 

13. 4P 

19. its¥ 

25. 

6 7 5 

2 0 0 

2. 

7 2 

TU 

8. 

124 

1/1 2 6 3 

I 4 * 16 

20 . mb 

26. 

990 

SOU 

3. 

V 

9. 

263 

TT 

1C 224 

21. VW 

27. 


4. 

4 8 

s 

10. 

328 

T2 

Ifi I-® 3 

lb. YU 

22. 

28. 

8000 
4 0 0 

5. 

53 

7 

11. 

2 6 4 

T2 

17 248 

17 * TOO 

00 5625 

1000 

29. 

4725 
- 25 ' 

6. 

9 1 

U 

12. 

3 6 6 

TS - 

•,0 3 8 7 

18 - ToTT 

O/i 4375 

24- 10 0 0 

30. 

3 125 
1 2U 


179. To reduce a mixed number to an improper fraction. 

Reduce 4| to an improper fraction. 

45 There are f in one unit. In 4 units, or 4, there are 4 times f, or V; 
— V + f = V. Therefore, 4| = V- 

In practice, the solution is as given in the margin, viz., multiply the 
5 whole number by the denominator of the fraction, and to the product 
77 add the numerator. Under the sum write the denominator of the 
8 fraction. 

EXERCISE 


180. Reduce the following mixed numbers to improper frac¬ 
tions (mentally, when possible): 


1 . 

3§ 

4. 9f 

7. 10} 

10 . 9} 

13. 72} 

2. 

41 

5. 6f 

8. 

11 . 17} 

14. 96} 

3. 

81 

6. 91 

9. 6J 

12. 19} 

15. 82} 



INTERCHANGE OF FRACTIONAL FORMS 


85 


16 . 855 

19. 38yf 

22. 236f 

25. 437f 

28. 322f 

17 . 45| 

20. 87i 

23. 328f 

26. 927f 

29. 972^ 

18. 19-rr 

21. 128f 

24. 482f 

27. 726f 

30. 1846f 


In the following pages both ways of writing fractions will be 
used. The student must be able to handle either form alone or 
both together. 

As illustrated in the chapter on Aliquot Parts, the common 
fraction is the more convenient form to use in many instances. 
In other cases, the decimal form is better. 


INTERCHANGE OF FRACTIONAL FORMS 

181. The student already knows that i of a dollar = 50 cents = 
.50 of a dollar. If this were not known, it could be determined 
thus: One dollar is written SI. 00. To find \ of any number, 
divide by 2. 

S.50 This simple example illustrates the method of changing 
2 ) S1.00 any common fraction to its equivalent decimal form, viz., 
Annex zeros to the numerator and divide by the denominator. 


EXERCISE 


182. Change the following fractions to their equivalent decimal 
form. (Find results mentally, when possible.) 


1* THT 

C 4 * 

TS 

9. IT 

13. TWO' 

17. if 

2. | 

6 * TS 

10. * 

14. if 

18. -gw 

3. A 

7. -5T 

IL 81) 

15. ff 

19* 

4. ITS 

8. f 

12. A 

16. TT7T 

20. ff 

*A common 

fraction in its lowest terms having 

a denominator that contains 


any factor other than 2 or 6 cannot be changed to a pure decimal. In such 
cases the division should be carried out two places and the fraction retained. 

.26f Hence, T V = -26| 

15)4.00 
30 
100 
90 

10 2 
15 "3 


Thus, 



86 


FRACTIONS 


If it were not known that 25 £ = .25 of a dollar = £ of a dollar, that fact 
could be determined by comparing 25ff with 100^ (100j£ being equal to one 
dollar). 25£ is T V* of 100^. 

5 | tVV = 5 | * = Hence, .25 = 

Observe that the decimal is written in the common fraction form by supply¬ 
ing its denominator, and reducing the resulting fraction to its lowest terms. 


183. The denominator of the decimal is 1 , with as many ciphers 
after it as there are places in the decimal. 

Change .461 to its common fraction form in its lowest terms. 


46fX3 

100X3 


= 10 


• 46 *=ioo 


140 14 _ 7_ 

300“ 2 30“l5’ 


.46f 

140 7 

or-= —• 

300 15 


.46f is a complex decimal. Its denominator is 100, giving a complex common 
46 f 

fraction of By the Principles of Division (No. 21, page 117), multiplying 


both dividend and divisor (numerator and denominator) by the same number 
does not change the value of the quotient. Hence, multiply both numerator 
and denominator by 3 (the denominator of the fraction f). The result is 
which equals tt, the desired result. 


EXERCISE 


184. Change the following decimals to their equivalent com¬ 
mon fraction form in their lowest terms. Determine results 
mentally, when possible. 


l. .8 

7. .032 

13. .003i 

19. .056 

2 . .24 

8 . .00125 

14. .1875 

20 . .104 

3. .56 

9. .0005 

15. .09f 

21. .00011 

4. .625 

10. .012J 

16. .09xt 

22 . .00031 

5. .711 

ll. .006i 

17. .07| 

23. .0142y 

6. .888 

12. ,002| 

is. .0015 

24. .0061 



LEAST COMMON DENOMINATOR 


185. What is the denominator of each of the following fractions: 


5 11 9 13 1 3 7 159 

TTT> T6> TIT) T5> T6> TT5"> TTT> TS - - 


Because their denominators are all alike, the fractions are said 
to have a common denominator. They are also called similar 
fractions. 





LEAST COMMON DENOMINATOR 


87 


186. Compare these denominators with each other: *, f, *, *, ■&, 
**, t> -18, .232. 

They are unlike, or dissimilar. 

What are dissimilar fractions? 

187. Fractions must be made similar before they can be added 
(See Principles of Addition, No. 5 , page 116.) 

188. What is the least common multiple of 2 , 4, 8 , and 16? 

189. 1 . Change *, f, f, and to similar fractions, having the 
smallest denominator possible. 

If the fractions are to be similar, the denominators must be 
alike. The least common denominator is the same as the least 
common multiple of the denominators. The least common de¬ 
nominator can be determined mentally. (See page 73 for find¬ 
ing L. C. M.) It is 16. 

As explained on page 82, reduce each of the fractions to 16ths. (State the 
principle of division that applies here.) 

2 . Reduce f, **, . 12 *, .33*, and .25 to similar fractions, having 
their least common denominator. 

Change the fractions in decimal form to their equiva¬ 
lent common fraction form, in their lowest terms. By 
inspection, it is seen that 24 is the least common denomi¬ 
nator. Each fraction is now changed to a fraction hav¬ 
ing 24 for its denominator. 

EXERCISE 


5 _ 1 5 

8 - 24 

11 _ 1 1 

24 ~ 24 

•12* = Wt 
QOl — 1 _ 8 

.oo 3 — 3 — 24 

•25 = i=& 


nr nr 


190. Reduce the following fractions to similar fractions having 
their least common denominator. (Do the work mentally when 
possible, using pencil only to tabulate results.) 


1 . 

12 3 

2 ) ¥> 4 

3. 

3 

4 ) 

2 . 

TO * 

4. 


9. 

TO TO -02* 



10 . 

8 2 5 

¥> ¥> 18 



11 . 

5 3 7 2 5 

¥> TO ¥> ¥¥ 


12 , 

5 1 OK 3 . 

' 16 , •J-ZO, 4 , 

.18 

f 

13. 

3 C 7 I 1 n 

T> O' 2) 2) 

,.25 

14. 

17 5 qql 

Tit ¥> 

.16 

2 

¥ 


5. f, .06*, .12* 


7 5 2 1 

'• ¥> ¥> nr 


6 . i, .9, .375 8 . I, f, I 

15. 1 %, *, .08*, .33* 

15* i) tq> i> i> T5T 

17. .25, .625, .06*, .5 


19. f, .31*, *, .16*, 

Ort 3 5 7 18 2 

20* TT> ¥> 3¥> TO ¥ 


88 


FRACTIONS 


ADDITION OF FRACTIONS 


191. 1 qt.+2 qt.+3 qt. = how many quarts? 

What kind of numbers can be added? 

1 fourth+3 fourths+2 fourths+3 fourths = how many fourths! 

m+f+l=? 


What kind of fractions can be added? 


192. 1 . Add f, i, f, i #. 

5 , 7 i 3 iii 2 _ is_ni Since these are similar fractions, they 

4 * may be added by taking the sum of their 
numerators, which is 18, and writing it over the common denominator, 8. 
V 8 , reduced to a mixed number, = 2\. 

2 . Find the sum of f, f, f, and hi- 

Before these fractions can be added they must be 
changed to similar fractions. By inspection, the least 
common denominator is seen to be 24. Instead of writing 
the 24 for each fraction, write it where it is needed, under 
the sum of the new numerators, as shown in the solution. 
Now find the numerators, writing each one after the frac- 
This arrangement makes the addition of the numerators 
very easy and quick, their sum being written, as shown, over the common 
denominator, 24; = 2ff, the desired result. 


2 _ 16 
sr — 

i= 18 

5 __ 15 
¥ — 

1 1 _ 22 
12 — _ 

ft- 2ft 

tion it represents 


3. Find the sum of 18f, 33f, and 68§. 

Write the integers and fractions, slightly separated, 

as shown in the solution. Add the fractions first by 
the method already explained. Their sum is 1$$. 
Add the integers, carrying the 1 from the sum of the 
120ff •§“§•= Iff fractions. The total result is 120f$. 

4 . Find the sum of 13f, 18.25, 64.345, and 56y. 


18 f= 20 
33 f= 24 
68 f = 15 


These fractions are best added in their 
decimal form. The first thing to do is to 
change each common fraction to its decimal 
form. Next write the numbers as shown, with 
the decimal points under one another. In 
changing £ and \ to their decimal form, the 
student should observe that the decimals are 
carried out as many places as there are places in the longest decimal in any of 
the other numbers. Thus, the longest decimal given is .345, having 3 places. 
Therefore, \ and -f are changed to 3-place decimals. This being done, the 
remaining common fractions, § and f, are parts of like orders and can be 
added. Add the common fractions as before, and carry the integral part as 
in the preceding example. 


13.333f= 7 
18.25 
64.345 
56.142f = 18 
152.071^4- ff = l^r 




ADDITION OF FRACTIONS 


89 


5. How many yards are there in 5 pieces of silk as follows: 41 2 , 
43 1 , 44, 423, 471? 

41 2 

431 The small numbers written a little above and to the right of the 
44 numbers 41, 43, etc., indicate quarter yards. That is, 41 2 yd. is 41 f 
423 yd.; 43 1 yd. is 43| yd., etc. This method of writing quarters is much 
used in the dry-goods business. These being similar fractions, the 
addition is made as already explained. 

218 3 


EXERCISE 


193. Add: 

!• i> I 

2* iy f 

3. f, h -h, i 

A 3 1 5 1 

ID 2; T2> TT 

5. 181, 16f, 17| 

6. 24f, 31f, 47f 

7 . 14J, 18.125, 27.5 

8. 1261, 162.5, 204.475 
9 I, 21, 13.324 

10. f, .4, I, .875 


U 4 7 

• ~Si s 

12 . h -h 

13. 18 1 , 17 2 , 16 s , 19 2 , 22' 
14. 48 [ , 72 s , 64 2 , 89 2 , 94 1 

15. 75f, 811, 63.427 

16. 41f, 72.325, 47.181 

17. 32f, 9If, 52.135 

18. 46 >, 72«, 84 1 , 92, 87 s 

19. 191, 181, 82f, 76f 

20. 211, 76f, 91f, 4.847 


MENTAL ADDITION OF FRACTIONS 

194. l. Find the sum of 1 and 1 

Observe that, in changing the fractions to similar fractions, the 
1 = Tif numerators of the similar fractions are the same numbers as the 
l=Tir denominators of the given fractions, and that the common 
~ denominator is the product of the same numbers, 3 and 4. There- 

12 fore, to find the sum of two fractions having 1 for a numerator, 

take the sum of the denominators for the numerator, and the product of the 
denominators for the denominator. 

2. Add i, |, 

Notice that ihe sum of the first two fractions, £ and 
i+ 8 24 i, i s a fraction (H) similar to the third fraction, inr, 

M+'2"! = I which may then be added mentally. 

The student should be always on the alert, looking for short methods. 


90 


FRACTIONS 


195. Add the following: 


1. 

2. 

3. 

4. 

5. 

6 . 

7 

8 . 

9 . 

10. 

¥ 

¥ 

\ 

l 

8 

t 

i 

¥ 

i 

* 

TT 

TT 

i 

T 

¥ 

1 

3 

i 

l 

T 

i 


T 

i 

HI. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

l 

10 

i 

l 

8 

l 

9 

TT 

l 

TT 

l 

8 

1 

T 

l 

6 

l 

T 

l 

l 

1 

1 

1 

l 

1 

1 

1 

l 

TT 

T 

T 

4 

T 

TT 

1 1" 

TT 

TT 

5 

21. 

22. 

23. 

24. 

25. 

26. 

27. 

28. 

29. 

30. 

l 

l 

l 

l 

X 

l 

l 

l 

l 

l 

2 

4 

3 

T 

¥ 

8 

¥ 

To 

¥ 

Y 

1 

3 

t 

1 

4 

l 

4 

1 

2 

1 

3 

l 

¥ 

1 

3 

1 

T 

¥ 

5 

9 

5 

3 

1 

1 1 

5 

4 

8 

5 

¥ 

To 

■6 

TT 

¥ 

12 

TT 

TT 

TT 

TT 

196. 

Find the sum of 

| and f. 







3 12 
5 1 7 

_24 
— ¥T 

2 

5 

is 2 times f , 

, and f 

is 2 times f . 

The sum of 

f and 


><2 = 24 ^ is if. Therefore, the sum of f and f is 2 times if, 

or if. That is, take 2 times the sum of the denomi¬ 


nators for the numerator of the sum. The denominator is the product of the 
denominators, the same as before. 


197. Add the following: 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

T 

T 

¥ 

¥ 

7 

T 

¥ 

7 

T 

T 

T 

T 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

6 

8 

T 

9 

¥ 

¥ 

¥ 

TT 

TT 

T 

¥ 

¥ 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

4 

4 

£ 

5 

6 . 

6 

2 

2 

2 

2 

3 

4 

5 

9 

8 

¥ 

T 

T 

¥ 

T 

T 

¥ 

4 

T 

4 

7 

TT 

5 

¥ 

5 

T 

6 

¥ 

TO - 

2 

7 

8 

2 T 

2 

¥ 

4 

TT 

2 

¥ 

1 

"TT 

2 

T 

11 

1 5 

t 

7 

¥¥ 

4 

7 

_8_ 
3 5 


198. The student should drill carefully and thoroughly on addi¬ 
tions like the following, till he can add simple fractions as readily 
as whole numbers. 

Add §, f-, f, and i%. 

To add these fractions, the student should “think” all opera¬ 
tions, naming only the results (“Think” the part in italics). 
The least common denominator is 16; the first numerator is 8; the 
second is 12, 20; the third is 10; 30, 39, tI = 2tV- 


SUBTRACTION OF FRACTIONS 


91 


199. Add the following: (mentally) 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

1 

¥ 

§ 

4 

t 

4 

¥ 

1 

2 

l 

¥ 

5 

¥ 


TT 

7 

TT 

1 

2 

3 

4 


1 

T¥ 

1 

T 

2 

¥ 

3 

4 

7 

T¥ 

2 

¥ 

3 

4 

1 

2 

5 

T¥ 

i 

4 


£ 

3 

4 

3 

4 

7 

TT 

3 

4 

1 

2 

7 

¥ 

5 

¥ 

7 

¥ 

A 

f 

A 

l 

1 0 

20 

7 

T2 

5 

¥ 

1 

2 

5 

T¥ 

3 

¥ 

2 

¥ 

f 







1 

2 

5 

8 

1 

¥ 


3 

T 

i 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

_! 

5 

5 

5 

9 

1 5 

5 

9 

i l 

i i 

2 

5 

¥ 

¥ 

8 

T¥ 

To 

¥T 

T¥ 

T¥ 

T5 

TT 

¥ 

9 

1 

1 

1 

3 

l 

1 

1 

2 

2 

3 

3 

2 

2 

2 

2 

¥ 

2 

nr 

2 

¥ 

¥ 

T 

T 

¥ 

5 

5 

3 

9 

2 

7 

2 

3 

5 

7 

5 

3 

¥ 

¥ 

4 

T¥ 

¥ 

¥ 

¥ 

4 

¥ 

¥ 

¥ 

4 

2 

2 

1 

1 

4 

9 

5 

1 

2 

1 

11 

1 1 

¥ 

1J 

4 

2 

5 

¥T 

¥ 

T 

¥ 

T 

TT 

TT 

1 

"S’ 

T 

1 

8 

f 

7 

To 

3 

4 

TT 

l 

¥ 

1 

¥ 

T4 

1 

2 

1 

2 


200. Add (mentally, when possible): 


1. *+*+♦+* 

2. 5f+i+J+^ 

3- f+T6 + -25.+ .37J 

4. 16f+17|-|-24|+56.5 

5. 47f+96.2+143^o+54.25 


6 . 122.125i+63J+88.0625+5H 

7. 49f+72f+99H+J 

8. 51 1 +64 2 +48 3 +54+50 3 

9. 32^+29|+19f+54 

10. 71.125+31.5+83.333i+.66| 


11-30. For further drill add the fractions on page 87. 


SUBTRACTION OF FRACTIONS 

201. l. 5 yd. minus 3 yd. =? 

2. 5 eighths minus 3 eighths = ? 

5 3 _ ? 

s s— • 

3 . What kind of fractions can be subtracted? 

4. From * take t+. 

9 3 _ 6 

TT IT TT* 

Since the fractions are similar, it is necessary only to take the difference be¬ 
tween the numerators, 9 and 3, which = 6, and under the 6 to write the com¬ 
mon denominator, thus, -rr- 


92 


FRACTIONS 


202. Find the difference between: 


1. 

| and 

3 

¥ 

s. 

tt and 

9. 1 and | 

2. 

t 9 t and 

7 

TT 

6. 

T 2 and yt 

10. .8 and .5 

3. 

| and 

l 

¥ 

7. 

1 and f 

11. .11 and .09 

4. 

f and 

2 

T 

8. 

T¥ and r6 

12. .50 and .25 


203. 1. From f take !• 

f = 15 Reduce the fractions to similar fractions, as in addition. In- 
1 = 8 stead of adding the new numerators, subtract the one from the 
7 other, which gives the result, 

2 4 

2. From 151 take 9f. 

14 4 _ ie | is larger than |. First, therefore, “borrow” 1 from 

g ’ 3 ._ 9 the 15, and add it to the 1 expressing the result as an 

-’ 4 — improper fraction, f. Now proceed as before: Change to 

T 2 similar fractions, and subtract, obtaining 5 T V as the result. 


204. Find the value of: 


1. 

7 

¥“ 

’f 

11. 

14*- 

7.331 

21. 

18.127 

-12.33f 

2. 

9 

TO 

5 

6 

12. 

381- 

21 H 

22. 

26.872 


3. 

3 

4 

2 

'¥ 

13. 

56*- 

29.5 

23. 

39.458 

i~ 18.12, 

4. 

T¥ 

“1 

14. 

76*' 

-67.18f 

24. 

79 t 3 t - 

18f 

5. 



15. 

98.16 

f-21.135 

25. 

711-281 

6. 

5f 

-3f 

16. 

88.124-571 

26. 

82 t 9 o — 

43i 

7. 

5i 

-2.75 

17. 

74*-- 

-28H 

27. 

67A" 

5 If 

8. 

18- 

1-12.161 

18. 

58*- 

27.9 

28. 

54.128 

-27.063 

9. 

24. 

831-181 

19. 

73*- 

34x4 

29. 

57.451 

-33.128 

10. 

63 

1-12.061 

20. 

841- 

72.13 

30. 

99f-74.25 


Mental Subtraction of Fractions 

205. l. From J take 1. 

In changing the fractions to similar fractions, notice that the 
numerators of the similar fractions are the same numbers as the 
denominators of the given fractions. The difference is £. 
Therefore,'take the difference between the denominators for the 
required numerator, and the product of the denominators for the required 
denominator. 




MULTIPLICATION OF FRACTIONS 


93 


2. From -§ take f. 

2. = i_o. The common denominator is 15. Observe that the new 
q _ _6 _ numerators are just twice the denominators of the given 
^ IA fractions, and that their difference, 4, is just twice the dif- 
T7> ference between the denominators 5 and 3. Therefore, take 
twice the difference between the denominators of the given fractions and write 
it over the common denominator. 

206. Subtract (mentally): 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

! 

l 

l 

l 

j 

' l 

i 

l 

l 

l 

1 

l 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

1 

l 

l 

l 

1 

l 

1 

1 

1 

1 

1 

1 

4 

3- 

T 

8 

3 

TO 

3 

3 

T 

8 

9 

10 


13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24 

i 

l 

i 

l 

l 

l 

l 

l 

l 

2 

2 

2 

4 

4 

3 

5 

3 

3 

5 

3 

3 

3 

3 

3 

1 

1 

1 

1 

1 

l 

1 

1 

l 

2 

2 

2 

1 1 

T2 

3 

T 

8 

9 

10 

Tl 

T2 

7 

3 

T 


207. Find (mentally, when possible) the value of: 


1. 

1 

2 

+ 

1 

3 

1 

“ 6 

11. 

1 

2 

+ 

2 

3 “ 

5 

“ 3 

21. 

7 

3 “ 

Ill. 
2 i 4 

3 

3 + 

T3 

2. 

1 

3 

+ 

1 

4 

5 

“ T2 

12. 

2 

3 

+ 

1 

4 

5 

“ 12 

22. 

J-Q ~ 

117 

~ 2 T“ 3 

— -13 

+ 3 

3. 

1 

3 

+ 

1 

5 

7 

1 5 

13. 

3 

4 

+ 

1 

3 “ 

7 

1 2 

23. 

1 + 

1 1 

3 2 

-* + 

2 

3 

4. 

2 

3 

+ 

2 

7 

5 

“21 

14. 

3 

5 

+ 

1 

2 

9 

“ T3 

24. 

* + 

2 5 1 1 

3“3+2 

1 

3 

5. 

3 

4 

+ 

-3 

3 

7 

“ 23 

15. 

3 

4 

+ 

3 

8 

1 

2 

25. 

1 + 

4 r 2 

3 

8 

1 

2 

6. 

2 

T 

+ 

2 

3 

1 

" 3 

16. 

5 

3 

+ 

1 

2 

2 

“ 3 

26. 

1 5 
T3 

l l 

"2 4 

+ 1 + 1 

7. 

1 

3 

+ 

3 

8 

" T2 

17. 

3 

8 

+ 

2 

3 

“ T2 

27. 

9 

T3 

_ 4 _l_ 1 
5 12 

+ i - 

_ 1 

21 

8. 

1 

4 

+ 

1 

3 

“ TO 

18. 

4 

3 

+ 

1 

3 

1 1 

“ T3 

28. 

8 

3 

1 1 4_ 2 _ 

3 3 i T3 

1 

4 

9. 

1 

8 

+ 

3 “ 

5 

“ 34 

19. 

5 

8 

+ 

3 _ 

4 

7 

“ 8 

29. 

11 _ 
12 

_ 3. _L l. 
4 3^ 6 

+ *- 

1 

2 

10. 

1 

4 

+ 

l 

T 

5 

“ TT 

20. 

3 

5 

+ 

3 _ 

7 

9 

" 33 

30. 

11 11 _ 5 

T3T 4 8 

+ *- 

_ 3 

T 


MULTIPLICATION OF FRACTIONS 

208. To multiply a whole number by a fraction or a fraction by 
a whole number. 

1. 4 times 3 gallons = how many gallons? 

2. 4 times 3 fifths = how many fifths? 4 X f = ? 

3. Does 4Xf = |X4? (Principle 13, page 117.) 


94 


FRACTIONS 


4. Compare these two solutions. 

5Xf=V R = 2f fX5= y = 2y. 

(X may be read “times,” “multiplied by,” or “of.” The first 
of these two equations may be read “5 times y=y,” or “5 multi¬ 
plied by f= y,” the second may be read “y of 5 = y,” or “f multi¬ 
plied by 5 = y ” It is not proper to say “5 of y,” or “y times 5.”.) 

State the principle of division that applies in these equations 
See page 117. 


5. Compare these two solutions: 


7V 5 _ 3 5 _ 5 _Ol 

( AtT-14- 2-^2- 


Observe that, in the first solution, the numerator of the fraction is multiplied 
by 7, and that the resulting fraction is reduced to its lowest terms by dividing 
both terms of the fraction by 7. 

In the second solution, notice that by dividing the denominator of the 
fraction, 14, by the multiplier, 7, the result, is obtained at once, without the 
intermediate step of multiplying by 7 (Principles 18, 19, page 117). 

To the Student. Study each problem carefully and determine the shortest 
solution. As far as possible, make all computations mentally, using pen or 
pencil to tabulate results. 

209. Find the value of: 


1. 

5 

X 

7 

¥ 

10. 

5 X i 

9 

0 

19. 

i l 
T2 

X 11 

oo 

M 

5 

T¥ 

of 

38 

2. 

8 

X 

3 

T 

ii. 

6 

X 

5 

6 

20. 

5 

¥ 

X 15 

29. 

11 

1 2 

of 

66 

3. 

9 

X 

7 

9 

12. 

8 

X 

9 

T6 

21. 

8 

¥ 

X 24 

30. 

7 

8 

of 

84 

4. 

6 

X 

3 

¥ 

13. 

5 

¥ 

X 

16 

22. 

9 

¥ 

X 24 

31. 

1 5 
T¥ 

of 

5 

5. 

5 

X 

4 

T 

14. 

7 

TO 

X 

15 

23. 

7 

T¥ 

X 60 

32. 

1 9 
2¥ 

of 

50 

6. 

9 

X 

8 

1 1 

15. 

7 

¥ 

X 

27 

24. 

8 

T¥ 

X 40 

33. 

2 1 
¥¥ 

of 

66 

7. 

3 

X 

2 

7 

16. 

1 5 

1 6 

X 

48 

25. 

f of 148 

34. 

1 2 
¥¥ 

of 

75 

8. 

4 

X 

8 

T¥ 

17. 

3 

4 

X 

64 

26. 

f of 123 

35. 

1 3 
20 

of 

48 

9. 

7 

X 

6 

TT 

18. 

8 

TT 

X 

55 

27. 

fof 72 

36. 

1 5 
T¥ 

of 

57 


210. In all cases in which the fraction is in its decimal form, pro¬ 
ceed in one of two ways: (1) If the decimal is one of the familiar 


MULTIPLICATION OF FRACTIONS 


95 


% 


aliquot or aliquant parte, use it in its common fraction form as 
illustrated in the chapter on Aliquot Parts. (2) In all other cases 
solve as follows: 


Multiply 425 by .217. 

425 

217 

1 * Perform the multiplication in the same manner as for simple num- 
2975 bers, and point off as many places in the product as there are places 
8925 in either or both the multiplier and multiplicand. 

92.225 


211. Multiply: 
l. 48 by .16| 

9. 368 by .649 

17. 

59 by .101 

2. 

93 by .123 

10 . 487 by .1245 

18. 

84 by .0125 

3. 

133 by .416 

11. 696 by .875 

19. 

96 by .125 

4. 

216 by .75 

12. 389 by .0625 

20. 

88 by .09 

5. 

874 by .613 

13. 496 by .083 

21. 

138 by .0001 

6. 

838 by .456 

14. 979 by .005 

22. 4585 by .01 

7. 

971 by .375 

15. 845 by .001 

23. 7345 by .003 

8. 

1240 by .625 

16. 859 by .0101 

24. 

250 by .004 


The student should observe carefully that on multiplying by 
such decimals as .1, .01, .001, .0001, and so on, the product con¬ 
tains the same figures as the multiplicand, with as many places 
pointed off as there are places in the decimal. Thus, multiply 4789 
by .001. The product is 4.789, the same order of digits, with 3 
places pointed off. 

Write the answers to the following: 


25. 8432 X.0001 

26. 9678 X.l 
26. 8439 X.001 

28. 9637 X.00001 

29. 45000X.01 


30. 64873X.00001 

31. 69478X.000001 

32. 83902X.01 

33. 84800X.001 

34. 12489 X.01 


35. 34876X.0001 

36. 48749X.01 

37. 87634X.001 

38. 85634X.l 

39. 7964 X.0001 


212. To find the product of an integer and a mixed number. 

1. At a yard, how much will 4 yd. of cloth cost? 

2 . At SI a yard, how much will 4 yd. of cloth cost? 

Compare the cost of 4 yd. at a yard and 4 yd. at $1 a yard, with the cost 

cf 4 yd. at $1£ a yard. 



96 


FRACTIONS 


3. At $3 a yard, how much will 1 a yard of silk cost? How 
much will 6 yd. cost? How much will 6J yd. cost? 

4 . How many are 6X41? 8X3J? 9X51? 31X8? 71X10? 
9^X4? 61X6? 

213. 1 . Multiply 864 by 13f. 

First multiply mentally by the fraction f; thus, | of 864 is 
108; 3 X 108 = 324. Next, multiply by the whole number, 13 , 
and add the results. 


864 

_13i 

324 

2592 

864 

11556 


2. Multiply 231f by 43. 

231| 

in 

_ First, multiply the fraction f by 43. If the integer, 43, is not 

3)86 exactly divisible by the denominator of the fraction, multiply 
28| the numerator of the fraction by 43; 43 X 2 = 86. Divide 86 
693 by 3, the denominator of the fraction, obtaining 28f. Then 
924 multiply 231 by 43, and add results. 

9961| 

3. Multiply 583 by 31.64f. 


31.64f 

583 

4)1749 

437i 

9492 

25312 

15820 

18450.491 


Note that the multiplicand, 583, is used as the multiplier 
for the purpose of shortening the work (Prin. 13, page 117). 
First multiply 583 by the numerator of the fraction, and 
divide the product by the denominator, as in the preceding 
example. Multiply 31.64 by 583 the same as though 31.64 
were a whole number, and add the results. In the product 
point off 2 places, for the 2 places in the number 31.64f. 


214. Find the 

1. 10 X 81 

2. 12 X 4f 

3 . 16 X 8f 

4 . 24 X 5f 


of: 

5 . 33 X Ilf 

6. 76 X 51 f 

7 . 21 X 7f 

8. 67 X 181 


9. 49 X 88| 

10. 83 X 31f 

11. 59 X 83f 

12. 38 X 17f 









MULTIPLICATION OF FRACTIONS 


97 


13 . 

16 # 

X 

28 

21 . 

91fff 

14 . 

18| 

X 

115 

22 . 

38* 

15 . 

14* 

X 

93 

23 . 

47* 

16 . 

15* 

X 

99 

24 . 

69* 

17 . 

24* 

X 

50 

25 . 

67 X 

18 . 

89** 

X 

160 

26 . 

87 X 

19 . 

39* 

X 

138 

27 . 

34 X 

20 . 

58* 

X 

44 

28 . 

39 X 


X 110 

29 . 98 X 74.661 

X 92 

30 . 38.13-?- X 45 

X 86 

31 . 56.42* X 66 

X 85 

32 . 3.123 X 86 

4.13f 

33. 71 X 16.10* 

•312i 

34 . 5.91* X 84 

4.37* 

35 . 8.31* X 34 

3.82J 

36 . 19.36* X 59 


215. To multiply a fraction by a fraction. 

1. How much is § of J? § of \? \ of J? 

2. If a yard of cloth costs f of a dollar, how 
much will J of a yard cost? 

216. l. What is i off? 



Let the circle represent a unit divided into thirds. 

If each.third is divided into four equal parts, the circle 
will be divided into 12 equal parts, or twelfths. f = tV 
i of |, then, is the same as f or t 8 if, which is equivalent 
to f of 8 parts, f of 8 parts is 2 parts; 3X2 parts is 
6 parts, or t 6 t F = f. Therefore, f of f = £. 

Instead of all the work indicated in this ex¬ 
planation, which has been given to make 
clear the principles involved, the solution may be made very 
short. Observe the following. 



By applying cancellation, the work of 
the solution is reduced to a minimum. 

Some problems do not admit of cancel¬ 
lation. For instance, in finding the value 
of f of l, cancellation cannot be applied. The solution consists 
in multiplying together the numerators for the numerator of the 
result, and multiplying together the denominators for the denomi¬ 
nator of the result. 


3 2 3X2 1 

4 ° 3 4X3~2' 

2 


The expression is equivalent to 


5X7 
6X8 = 


35 

48* 


VAN TUYL’S NEW COMP. AR.—7 





98 


FRACTIONS 


2. Multiply .43 by .032. 

Multiply as in whole numbers, pointing off as many places in 
the product as there are places in both multiplicand and mul¬ 
tiplier. 

.01376 


217. Find the products: 


1. 

2 

t 

of 

9 

TO 

10. 

¥ Of i Of t 

19. 

.4123 

X 

.814 

2. 

5 

8 

of 

4 

T 

11. 

TT of ■§■ Of y 

20. 

.963 

X 

.0001 

3. 

7 

¥ 

of 

9 

TT 

12. 

toftof* 

21. 

8.24 

X 

.001 

4. 

T 

of 

11 

TT 

13. 

«of-» of T % 

22. 

.4237 

X 

1.001 

5. 

3 

T 

of 

7 

8 

14. 

•44 of f of 4 

23. 

.954 

X 

10.11 

6. 

44 

of 

3 

T 

15. 

f of 4r of 44 

24. 

76.98 

X 

11.11 

7. 

TT 

of 

9 

TT 

16. 

•f of ¥ of TT 

25. 

8.329 

X 

8.96 

8. 

8 

¥ 

of 

44 

17. 

9 */vF 8 45 

TT 01 TT 01 TT 

26. 

4.73 

X 

100.10 

9. 

7 

TO 

of 

44 

18. 

7 J 24 15 

TT 01 35 OI XT 

27. 

89.48 

X 

3.824 


218. To multiply a mixed number by a fraction. 

1 . What is 4 of 4? \ of 4? Add the two answers. 

2. What is \ of 44? i of 64? i of 8i? 

3. What is | of 64? i of 124? 4 of 204? 


.43 

.032 

86 

129 


219. Find the value of 4 of 313|. 


7 4 7 

$ ° f 5 10 


£of 313 = H 8 - 1 =273| 

7 _ 28 
TT 

t 2 3 TT_ 12 3 
TXT 4 0 -*-4 0 


First, multiply the fractions together, ob¬ 
taining T V 

Next, take £ of 313, which gives 273£. 
(For explanation, see page 96.) 

Now add -nr and 273 £, and the desired 
result is obtained. 


220. Find the products: 


l. 

4X181 

5. f X91y 

9. 

f X148J 

2. 

yX33^- 

6. 4X88y 

10. 

IX131f 

3. 

Axief 

7. i 9 xX 84 t V 

11. 

48JX-,\ 

4. 

£X39£ 

8. x4X127f 

12. 

68£Xf 




MULTIPLICATION OF FRACTIONS 99 


13. 

137} 

X 

6 

T 

19. 

591 

4 w 4 

5 A 3" 

25. 

3 

T 

X 

414| 

14 . 

246} 

X 

5 

20. 

563} X } 

26. 

7 

¥ 

X 

214} 

15. 

337} 

X 

3 

T 

21 . 

3 

Tcf 

X 416} 

27. 

3 

8 

X 

328^ 

16. 

498} 

X 

2 

S 

22. 

5 

9 

X 382} 

28. 

4 

3- 

X 

564f 

17 . 

468}f 

X 

3 

4 

23. 

7 

TT 

X438| 

29. 

7 

ITT 

X 

366} 

18 . 

483 t 3 t 

X 

4 

24 . 

5 

•S' 

X 938} 

30. 

9 

TO 

X 

455} 


221. To multiply a mixed number by a mixed number. 


Find the value of 31} X17|. 


31} 

17f 
U> 15 
19}, 12 
12}, 24 

527_ 

559}-} }} = 1-}} 

(4) 17 X 31 = 527. 


This solution is what 
is known as the “four- 
step process” of multi¬ 
plying mixed numbers. 
For clearness, the four 
steps are written out in full. The student should 
observe these four steps very carefully: (1) f of f = 
If, which is written as a part of the product. (2) 
f of 31 = 19f, which is written as in the solution. 
(3) 17 X f = 12f, which is written under the 19f. 

Add the several results. The final result is 559|f. 


i 5 3 _ 1 5 

1. 8 01 T 

2. f of 31 = 19} 

3 . 17Xf = 121 

4 . 17X31 = 527 


Multiply 29} by 15}. 

29} 1. } of J = r6 

15j f}of29l 

11A Uof 15/ 

435 \ of 44 = 11 

446A 3 . 15X29 = 435 


It is frequently possible to shorten the 
“four-step process” as illustrated in this 
exercise. In all cases in which the frac¬ 
tions are alike, the integral part of the 
multiplier and the multiplicand may be 
added and their sum multiplied by one 
of the fractions. “Steps” 2 and 3 are 
combined into one “step.” 


222. Multiply: 

1. 15f X 16} 

2. 18} X 26} 

3 . 27} X 34} 

4 . 37} X 29} 

5 . 38} X 34} 


6. 56} X 9} 

7 . 4} X 9f 

8. 76} X 12} 

9 . 894- X 171 

10. 93? X 27? 


11. 87| X 29i 

12. 91J X 41y 

13 . 471 x 38f 

14 . 93? X 86f 

15 . 74? X 19| 







100 


FRACTIONS 


16. 88fX55f 

17. 1.341 X.19f 

18. 2.28| X.67# 
is. 2.14AX.48} 


20. ,246|X.96§ 

21. 3.24f X.56| 

22. 6.32| X.16# 

23. 4.56f X.59| 


24. 5.98fX 1.22| 

25. 8.46f X1.38f 

26. 9.66|X 1.44f 

27. 8.62§X2.33£ 


Find the total value of each of the following: 

Fractions, when not expressed in full, are to be considered as fourths. 
Thus, $.07*=*.07f. 


28. 


29. 

30. 


273 yd. at $.07 2 . 

351 yd. at $.071. 

641 yd. at $.082. 

31 1 yd. at 

.083. 

433 yd. at .081. 

76 2 yd. at 

.073. 

45 2 yd. at 

.09h 

72iyd. at .05 2 . 

48 3 yd. at 

.082. 

56 3 yd. at 

.073. 

533 yd. at .06 2 . 

39 2 yd. at 

.043. 

38 2 yd. at 

.061. 

58 2 yd. at .04 s . 

17 2 yd. at 

.123. 

51 2 yd. at 

.073. 

621yd. at .071. 

19 3 yd. at 

.133. 

31. 


32. 

33. 

723 y d. at $.10h 

213 yd. at $. 072 . 

44i yd. at $.081. 

42 2 yd. at 

.092. 

343 yd. at .06 3 . 

29 3 yd. at 

.071. 

63 1 yd. at 

.072. 

56 2 yd. at .08 2 . 

64 2 yd. at 

.073. 

16 3 yd. at 

.152. 

47iyd. at .091. 

29 3 yd. at 

.052. 

22 2 yd. at 

.183. 

533 yd. at .073. 

12 2 yd. at 

.261. 

31 3 yd. at 

.232. 

291yd. at .071. 

14 3 yd. at 

.352. 

34. 


35. 

36. 


592 yd. at $.052. 

261 yd. at $.072. 

322 y d. a t $,06 2 . 

39 3 yd. at 

.093. 

323 yd. at .072. 

54 1 yd. at 

.071. 

45 1 yd. at 

.062. 

492 yd. at .08 3 . 

69 3 yd. at 

•093. 

73 2 yd. at 

.08?. 

382 yd. at .222. 

57 1 yd. at 

.071. 

67 3 yd. at 

.072. 

593 yd. at .332. 

48 2 yd. at 

062. 

83 1 yd at 

.091. 

472 yd at .172. 

27 3 yd. at 

.053. 


MULTIPLICATION OF FRACTIONS 


101 


Mental Multiplication of Mixed Numbers 

223. When each fraction is 


l. Multiply 6 J by 6 J. 


( 0 ) 6 j 

(b) 65 

6 ! 


3J 

42J 

3 


36 


42j 



In (a) the complete “four-step” solution (page 99) 
is given. Note that the sum of the two cross products 
(3 and 3) is 6 . Hence, instead of taking 6 X 6 , and 
adding 6 , add 1 to the multiplicand, and take 6X7, 
as in ( b ). 


2 . Multiply 9.5 by 9.5. 


9.5 

g 5 As in ( 6 ), example 1, .5 X .5 = .25. Add 1 to one of the 9’s and 
-- multiply by the other 9, and obtain 90. Hence, 9.5 X 9.5 = 90.25. 


224. Multiply (mentally): 


l. 


2 . 


3. 


5. 


8 J by 8 J 

6 . 1 | by 1 § 

11. 

5| by 5§ 

7* by 7* 

7. 10|byl0i 

12. 

6.5 by 6.5 

3.5 by 3.5 

8 . 11.5 by 11.5 

13. 

9? by 9i 

4.5 by 4.5 

9. 13§ by 13§ 

14. 

15§ by 15§ 

2 h by 2 i 

10 . 12.5 by 12.5 

15. 

20i by 201 


225. l. Multiply by 5|. 


7i 

At 

41i 


i of 7 + i of 5 = h of (7 + 5) = 6 . Take 5X7, and add the 6 , 
which gives 41 \ as the product. 


2 . Multiply 9§ by 4J. 

91 The sum of the integers (4 + 9 = 13) being an odd number, \ of 
41 the sum gives a mixed number, 6 |. § of \ = J, to which is added the 

—- i of the mixed number ( 6 |), giving f, the fractional part of the prod- 
uct. Now take 4 times 9, and add the 6 , obtaining 42f, the desired 
result. 

In problems like the last two examples, observe first whether the sum of 
the integers is even or odd. If even, write as the first part of the product 
.25, or 25, as the case may require; if odd, write f, .75, or 75. For the re¬ 
mainder of the product multiply the integers together, and add the integral 
part of half their sum. (Compare with Art. 107.) 



102 


FRACTIONS 


When the fractions are alike. 

3. Multiply 8| by 5f. 

8 } 

First multiply § by f. Next find f of 13, (8 + 5), which is 8 f. 
4 Then find 5X8, which is 40. It should be observed that although 
J § of 13 gives a mixed number, 8 f, the solution is shorter than by the 
8 } other method. Now there are only two fractions to be added, instead 
40 of three, as there would be if 8 and 5 were multiplied separately. 

m 

The student will be surprised at the rapidity with which he will 
be able to make multiplications like the above examples, after a 
little practice. 


226. Multiply (mentally): 


1. 

6 }X10} 

8. 

7}X8} 

15. 

15fX6f 

2. 

8}X7} 

9. 

10}X15} 

16. 

18|X7f 

3. 

5}X 11} 

10. 

11.5X8.5 

17. 

161X8} 

4. 

8.25X4.25 

11. 

14} X 8} 

18. 

llfX7f 

5. 

6}X9} 

12. 

5}X7} 

19. 

13}X2} 

6. 

7}X9} 

13. 

12fX9f 

20. 

15.75X1.75 

7. 

11.25X9.25 

14. 

14fX7f 

21. 

14yX4y 


227. Find the total value of each of the following groups: 


1 . 

65 crates at $6.50. 
85 crates at 7.50. 
4} bu. at .75. 
9} bu. at .85. 

4. 

5} yd. at $.07}. 

9} yd. at .18}. 

7} yd. at .09}. 
13} yd. at .07}. 


2. 

10} lb. at $.75. 
5} lb. at .65. 
7} lb. at .07}. 
3} lb. at .15} 

5. 

8} lb. at $.08}. 

9} lb. at .Ilf, 

5f lb. at 3.75. 

12} lb. at 4.75. 


12} A. at $125. 
15}A. at 155. 
16} A. at 85. 
20} A. at 75. 

6 . 


15} bu. at $1.50. 

350 lb. at 1.50 per cwt. 
2500 ft. at 18.50 per M. 
4500 ft. at 6.50 per M. 


PAY ROLLS 


103 


228. The 44 hour week (8 hr. daily from Monday to Friday 
inclusive, and 4 hr. on Saturday) is fast becoming the standard 
week of labor. 

The scale of wages is based on the hourly or daily rate. The 
day is thought of as beginning at 8 a.m. and ending at 5 p.m., 
with 1 hr. for luncheon, except Saturday, when the day ends at 

12 M. 

Hours of labor between 8 a.m. and 5 p.m. (or between any 
other regularly prescribed hours of beginning and ending the day 
of labor, as is done in certain kinds of business) is regular time. 
Hours of labor after 5 p m. (or other prescribed closing hour) is 
generally considered overtime. 

Overtime is usually paid for at a higher rate than regular time. 
The extra wage may be a quarter, or a half greater than, or even 
double, the regular wage. It is expressed as time and a quarter , 
time and a half , or double time. In terms of figures that means 
that a man who receives 60 an hour regular time would receive 
for 1 hour over time, 75 ^ if he was allowed time and a quarter 
(1JX60^ = 75 £); 90 fi if he was allowed time and a half (1|X60^ 
= 90 jzf), etc. 


The following illustration will show how an employee's weekly 
wages are determined. 



M 


w 

Th 



Hours 

Rate 

Wages 


T 

F 

s 

Reg. 

Time 

Over 

Time 

Reg. 

Time 

Over 

Time 

Total 

Anson, John 

8 

9 

10 

5 

7 . 

4 

40 

3 

60c 

24.00 

2.70 

26.70 


Regular time and overtime are reckoned separately. The 
regular time for Tuesday and Wednesday is 8 hr. for each day. 
Hence the total regular time is 8 hr.+8 hr.+8 hr.+5 hr.+7 hr. 
+4 hr. =40 hr. The overtime is 1 hr. on Tuesday and 2 hr. on 
Wednesday, or a total of 3 hr. The rate (wage per hour) being 
60 0, the regular wages amount to 40X60^ or $24.00. It is as¬ 
sumed, in this problem, that time and a half is allowed for over¬ 
time. Since the regular rate is 60^, the overtime rate is 90 i 
per hour. The overtime wages amount to 3X90 or $2.70. The 
total wages are $26.70. 





















104 


FRACTIONS 


In many kinds of business the money due an employee for 
wages is put into a “pay envelope” and handed to him at the 
close of the week’s work. The number and kinds of bills and 
coins required are determined by means of a “currency break¬ 
up” or change sheet. For the wages in the above problem, together 
with the wages of several other employees, the currency break-up 
would appear as follows: 


No. 

Amt. 

$20 

$10 

$5 

$2 

$i 

50c 

25c 

10c 

5c 

lc 

1. 

$26.70 

1 


1 


1 

1 


2 



2 . 

27.18 

1 


1 

1 




1 

1 

3 

3. 

31.23 

1 

1 



1 



2 


3 

4. 

29.87 

1 


1 

2 


1 

1 

1 


2 

5. 

34.69 

1 

1 


2 


1 


1 

1 

4 

149.67 

5 

2 

3 

5 

2 

3 

1 

7 

2 

12 



100 

20 

15 

10 

2 

1.50 

.25 

.70 

.10 

.12 


Having thus determined 
the correct amount of change 
required, the result is copied 
on a currency memorandum, 
and sent to the bank with a 
pay-roll check. In cashing 
the check the paying teller 
provides the exact amount 
of change called for. The 
payroll clerk is then able to 
fill the pay envelopes with¬ 
out delay. 


THE COLONIAL BANK 

CURRENCY MEMORANDUM 

New York, Mar. 22, 1924 
Depositor Robert Levitt 


* 

Bills $ 1 

Dollars 

Cts. 


2 


‘ 2 


10 


8 5 


15 


8 10 


20 


8 20 


100 


“ 50 




8 100 




Coin: Pennies 



12 

8 Nickels 



10 

8 Dimes 



70 

8 Quarters 



25 

8 Halves 


1 

50 

8 Dollars 




8 Gold 




Total 


149 

67 



























































































PAY ROLLS 


105 


No. / 73 


New York _^^^. 19 . 

The Williams Bank 








BL^THRO £ 




GH NEW YORK CLEARING HOUSE 




Pay-Roll Check 


229. l. The following time sheet is based on a 44-hour week. 
Allow time and a half for overtime. Using the form shown on 
page 103 find the total wages for each employee. Prepare the 
currency break-up, a currency memorandum, and write the pay¬ 
roll check. 


No. 

M 

T 

w 

Th 

F 

s 

Hours 


Wages 

Reg. 

Time 

Over 

Time 

Rate 

Reg. 

Time 

Over 

Time 

Total 

1 . 

8 

7 

74 

6 

9 

4 



.60 




2. 

7 

84 

6 

9 

9 

5 



.60 




3. 

9 

10 

7 

6 

74 

64 



.64 




4. 

8 

8 

7 

8 

8 

6 



.62 




5. 

9 

7 

6 

44 

8 

5 



.50 




6 . 

10 

8 

7 

9 

8 

4 



.60 




7. 

9 

7 

8 

6 

9 

4 



.50 




8 . 

84 

7 

6 

94 

81 

74 



.64 





2. In the following time sheet the rate is for an 8-hour day, 
(44-hour week) with time and a quarter for overtime. Prepare the 
payroll, currency memorandum and check, using the above forms. 


No. 

Names of Employees 

M 

T 

w 

Th 

F 

s 

Rate 

1 . 

Addison, James 

8 

10 

9 

11 

2 

4 

6.40 

2. 

Buell, Henry 

9 

6 

9 

81 

9 

4 

6.00 

3. 

Crane, John 

10 

94 

8 

7 

8 

3 

6.00 

4. 

Crawford, Wm. 

9 

9 

8 

6 

10 

4 

6.00 

5. 

Jones, Albert 

10 

8 

94 

9 

0 

4 

6.00 

6. 

Landis, Frank 

8 

4 * 

9 

8 

10 

4 

5.00 













































106 


FRACTION 


Find the total time and the wages for the week. Make no 
allowance for overtime. 


Form No. 3 

No.18. 

Name. William MacCormack . 

Total Time.Hr. Rate, 55?f. 

Total Wages.$. 


Week Ending. January 7 .19 


In 

Out 

In 

Out 

In 

Out 

7.50 

11.30 

12.20 

5.30 





12.30 

5.30 

6.45 

11.30 

7.45 

11.30 

12.45 

5.45 



7.55 

12.15 



6.10 

11.15 



12.00 

5.40 

6.30 

11.20 

8.00 

12.20 

12.50 

6.10 



8.30 

12.15 

1.30 

6.10 




4. A gang of carpenters working under the 8-hour law, with a 
maximum day of 10 hours, and one-half extra pay for overtime, 
worked one week as follows. 


Time Sheet. Week Ending March 22, 1924. 


Names 

M. 

T. 

w. 

Th. 

F. 

s. 

Total 

Time 

Rate 
per Day 

Bigelow, A. E. (foreman) 

8 

81 

8 

9 

8 

10 


$7.50 

Arnold, A. W. 

8 

8 

8 

8 

8 

9 


7.00 

Babcock, C. D. 

8 

8 

8 

9 

8 

8 


7.00 

Comstock, L. O. ... 

8 

8 

8 

81 

8 

9 


7.00 

Evans, M. R. 

8 

8 

8 

81 

8 

8 


7.00 

Geraghty, P. O. ... 

8 

81 

8 

81 

8 

9 


7.00 

Morrow, G. W. 

8 

81 

8 

8 

8 

8 


7.00 

Peters, A. B. 

8 

81 

8 

81 

8 

8 


6.00 

Rogers, H. F. 

8 

81 

8 

8 

8 

9 


6.00 

Simler, H. 

8 

8 

8 

8 

8 

8 


6.00 


Prepare a pay roll for the gang. 













































PIECE WORK—DIFFERENTIAL RATE PLAN 


107 


230. Instead of working by the day or the hour, some employ¬ 
ees are paid by the piece. In that case their wages vary in pro¬ 
portion to the amount of work done. To find the amount of daily 
or weekly wages is simply to multiply the price per piece by the 
number of pieces of work completed. 

In some factories the piecework plan is modified so that a 
rapid or skilled worker receives a higher price per piece than a 
slower or unskilled worker receives. It may be that it has been 
found from experience that the completion of from 32 to 35 
pieces is a standard or average day’s work, and that the 
manufacturer can afford to pay 15 ^ apiece for the labor. To 
provide a slow worker with tools and machinery, heat, light, 
and factory space in which to work, costs just as much as it does 
to provide for a fast worker. Assume that the overhead ex¬ 
pense averages $1 a day for each employee. Then if one worker 
completes 25 articles per day, the overhead expense per article 
is 4 jf; if he completes 33 articles per day, the overhead is 3 ^ 
each; and if he completes 40 articles per day, the overhead is but 
2\ ff each. 

Therefore, to equalize cost of production per article, a “differ¬ 
ential rate” is paid. A slow worker receives less, and a fast worker 
receives more, per piece. 

1. Using the following schedule of rates, find the daily wages 
and the total wages for the week, for each employee: 


Piecework Pay Roll Schedule op Rates 




Number op Pieces 



Price 

Names 







Pieces per Day 

per Piece 

M. 

T. 

w. 

Th. 

F. 

S.* 





20 to 23 

13 y 2 i 














Brown, John 

33 

31 

34 

28 

36 

20 

24 to 27 








28 to 31 

14 y 2 t 

Carey, Leon 

28 

30 

27 

32 

33 

18 

32 to 35 (Standard) 

IH 

Dunne, Theo. 

24 

26 

29 

27 

30 

15 

36 to 39 

1 5& 







40 to 43 

16*f 

Evers, Mark 

34 

36 

40 

38 

39 

21 

44 to 47 

16^ 


♦Four hours only. Use the rate for twice the number of pieces finished. 




















108 


FRACTIONS 


DIVISION OF FRACTIONS 


231. To divide a fraction by an integer. 

Into how many parts is this circle divided? What part of 
the circle is marked off by the curved line? If these yg are divided 
into 3 equal parts, how many sixteenths will 
there be in each part? 

9 . Q_3_ 

16 • 16‘ 

How does the fraction yg compare in value 
with the fraction T 9 e? Formulate a second way 
of dividing a fraction by an integer. (Prin. 17, 
page 117.) 



232. 1 . Divide f by 3. §-^3=^4. 

Multiply the denominator of the fraction by the integer, 3, and write the 
numerator, 5, over the product. This divides the fraction, because multiply¬ 
ing the divisor (denominator) divides the quotient (which, in this case is the 
fraction). (Prin. 17, page 117.) 


2. Divide yt by 2. = 

Divide the numerator of the fraction by the integer, 2, and write the result 
over the denominator, 11. This divides the fraction, because dividing the 
dividend (numerator) divides the quotient (the fraction). (Prin. 20, page 117.) 

This solution should always be used when the integer is an 
exact divisor of the numerator of the fraction. 


3. Find the value of .0344-^8. 


0043 The w h en the divisor is a whole number, as 

- in this case, is to place the point for the quotient. Place 

8).0344 it directly above the point in the dividend, as shown in the 
example. By dividing thu% “8 is contained in 0, 0 times; 
8 in 3, 0 times; 8 in 34, 4 times, etc.”; placing the figures of the quotient above 
the figures of the dividend, as here shown, the position of the decimal ooint in 
the quotient is always right. 

233. Find the value of: 


1 . f-J-3 5. .0125-^5 

2. f-i-4 6. .464 -Sr 8 

3. f-r-3 7. ^3 

4. ^-6 8 . *-3 


9. 

11 . q 

13. 

7 

20’ 

^-5 

10 . 

.968-7-4 

14. 

10 . Q 
1 7 " r ® 

11 . 

.8464-16 

15. 

1 1 
TS~ 

-9 

12 . 

.0035-^5 

16. 

24 . 
3 1 * 

r 6 





DIVISION OF FRACTIONS 109 


17. .3125-f-25 

23. 

£-4-5 

29. 

.0125 + 500 

35. ^ 104 

18. .738-^9 

24. 

? -5 

30. 

.8448 4- 8 

36. .00126 -i-63 

19. .001-MOO 

25. 

19 . Q 

~s • 

31. 

1 5 9 . Q 

1 60 

37 . .216 + 24 

20 . .00128-M 6 

26. 

3 1 _»_ ft 

- 52 —V 

32. 

249 . OQ 

32^ — 00 

38. .4888 + 8 

21. if-r-4 

27. 

43 _s_7 

33. 

8 9 6 . 1 OO 

39. .03036+11 

8 • * 

1024 * 

22. M-5 

28. 

.125-7-50 

34. 

+S$*-s-121 

40. .3069 + 11 


234. To divide a mixed number by an integer. 

If $1J is divided equally among 3 boys, how much will each 
receive? If 9 lb. of coffee cost $2f, what part of a dollar does 
1 lb. cost? 

1H3 = ? 2|+9 = ? 2§ + 8 = ? If+4 = ? 

235. 1. Divide 5f by 7. 

5f = 4 -j When the mixed number is small, change it to an improper 
^ -i- 7 = ff fraction, and divide as in dividing a fraction by an integer. 

2. Divide 3478f by 8. 

434f In problems like this, in which the mixed number is much 

o\ 04702. larger than the divisor, do not change the mixed number 
2 9 5 6 to an improper fraction. Divide the integral part of the 
9 dividend by the divisor as if there were no fraction in the 

+ 8 = f dividend. 8 is contained in 3478, 434 times, with a remainder 
of 6, which, with the fraction f, makes a total remainder of 6|. Dividing 6f by 
8 as explained in the preceding example gives the completed quotient, 434f. 

The student should divide 6f by 8, and all similar expressions, 
mentally. It is written out in full here for the sake of clearness. 

3. Divide 42.3 by 25. 

Place the point for the quotient directly above the point 
in the dividend. The important part, now, is to write the 
first figure of the quotient in the right place. Observe that 1, 
the first figure of the quotient, is written over the 2, the figure 
farthest to the right in the dividend, that is used to contain the 
divisor, 25. Continue the division, annexing ciphers in the divi¬ 
dend as needed, and placing the figures of the quotient over each 
succeeding figure in the dividend, respectively. 

Note. If the figures of the quotient are placed correctly, the 
last, or right-hand figure, of the quotient will stand directly 
above the last figure of the dividend. 


1.692 
25)42.300 
25 
17 3 
15 0 
2 30 
2 25 
50 
50 








110 


FRACTIONS 


236. Find the value of: 


l. 3J-5-5 

16. 3924f-M2 

31. 478.336-5-16 

2. 6§-5-10 

17. 4964| + 11 

32. 394.128-5-8 

3. 8J-5-7 

18. 68479 + 96 

33. 48.3744-5-24 

4. 3f-5-7 

19. 3478|+84 

34. 9674.88-5-48 

5. 9f-5-14 

20 . 9674J + 99 

35. 364.64-5-160 

6 . 181-5-8 

21 . 8496 t ’ t +64 

36. 511.375-5-125 

7. 26J-5-4 

22 . 4589/o+5 

37. 895.4-5-110 

8 . 47f-5-2 

23. 896tx + 17 

38. 6.4872-5-4 

9. 167f-5-21 

24 . 974fV+11 

39. 9.6741-5-8 

io. 568f-5-43 

25. 897£+15 

40. 4.592-5-16 

ll. 828 it-5- 69 

26. 969 t \+7 

41. 9.34^-5-12 

12 . 939 t \-5-22 

27. 896^ 3 o-5-8 

42. 6.431-5-5 

13. 896£-s-15 

28. 487f-5-10 

43. 6.38|-5-9 

14. 948f -5-12 

29. 389^2--5-6 

44. 9.72f-5-11 

15. 776J-5-9 

30. 9671-5-5 

45. 106.881-5-15 


237. To divide an integer, a fraction, or a mixed number by a 
fraction. 

How many times does the whole circle con¬ 
tain | of the circle? 

How many times does it contain \ of the 
circle? f of the circle? 

l-§ = ? l-i = ? l + i~? 

If \ is contained in 1, 2 times, how many 
times is J contained in 2? in 3? in 4? in 6? in 10? 



Observe, in the first equation, that the answer, 2, or its equiv¬ 
alent, f, is the fraction \ inverted; that is, the denominator is 
written for the numerator, and the numerator for the denominator. 

In the second equation, notice that the answer 6, or f, is 3 times 
as large as the answer in the first equation. That is, to divide by J 
is to multiply by its reciprocal, 2. 

In the same circle count off 3 eighths. How many times will 
these f be contained in the circle? 





DIVISION OF FRACTIONS 


111 


Call the “eighths” “parts,” and divide 8 parts (that is, the 
circle) by 3 parts. 8 parts3 parts = 2f times. That is, 1-J-f = 2f. 
But 2§ = -§, and -J is the fraction f inverted. 

The result of 1 divided by any fraction is that fraction inverted. 

238. The reciprocal of a fraction is the fraction inverted. 

239. The reciprocal of any number is 1 divided by that number. 
The reciprocal of 4 is J; of 8 is etc. 

What is the reciprocal of }? of |? of f ? of 10? of 12? 

How may an integer be divided by a fraction? 

To divide by a fraction , multiply by its reciprocal. 

1 . Divide 6J by £. 

5 



= 6fX: 


20 5_25_ 

i x rir 


To divide by f- is to multiply by its reciprocal, 



61 multiplied by i gives 


2 . Divide $ of f of f by \ of f of x %. 
*of t oft + f off 0fT% = 


-X^X-X-X-X —= —. 
0 $ 3 3 2 9 27 


In dividing by a compound fraction, invert all the separate fractions com¬ 
posing the divisor, and treat the problem as an exercise in cancellation. 

Note. A compound fraction is a fraction of a fraction: thus f of f ; i of f 
of etc. 


3. Divide 4.8325 by .025. 

193.3 When the divisor is a decimal, change it to an integer 
by moving its point to the right-hand side of the decimal. 
In this example the point is moved three places, which is 
equivalent to multiplying by 1000 . Since the divisor is 
233 multiplied by 1000 , the dividend must likewise be multi- 
225 plied by 1000 (Prin. 21, page 117). Therefore, the point in 
g2 the dividend is moved 3 places to the right. Before 
75 dividing, place the point for the quotient directly above the 

- point in the dividend after it is moved. 

^ The division is now performed in the same manner as in 
75 dividing by integers. 


X 025.)4 X 832.5 

25 




112 


FRACTIONS 


4. Divide 5000 by .0005. 

1000 0000. As in the preceding example, change the divisor 
non*; ^nnn nnnn to an integer by moving the point to the right-hand 
x *' x ' side of the decimal, that is, by multiplying by 10,000. 
The dividend also must be multiplied by 10,000; that is, the point must be 
moved 4 places to the right. In this example it is necessary to make the 
places by annexing ciphers. 

Remember, the point in the dividend must he moved in the same direction 
as many places as the point in the divisor is moved to make the divisor a whole 
number. Annex ciphers when necessary. 

Remember, also, to place the point for the quotient before dividing. 

The process of division is then the same as in simple numbers. 


240. Find the value of: 


1 . 

2 . 3 

14. 

28f-5-xV 

27. 404.04 —.20 

2 . 

4 . 9 

^5 ~T7T 

15. 

181-f 

28. 

96.848-r-1.60 

3. 

*+* 

16. 

.2862 -5- .009 

29. 

74.64 h- 1.25 

4. 

r. . 5 

7 * 9 

17. 

34.74-k018 

30. 

395.20 4- .0125 

5. 

11.2 

18. 

49.52-i-.16 

31. 

16 4-f 

6 . 

Ol . 4 
^4 * 5 

19. 

.004^ .0002 

32. 

284-y 

7. 


20 . 

.400^.004 

33. 

364-if 

8 . 

15KI 

21 . 

.800.0001 

34. 

894-| 

9. 

foff4-§off 

22 . 

.008-K0001 

35. 

564-f 

10 . 

f off-H A off 

23. 

.124 -.624 

36. 

384-^| 

11 . 

t of f-j-fr of It 

24. 

4.875 —.0625 

37. 

42 4- T ? 

12 . 28JXt%"^t% I - 

25. 

8.5 —.3125 

38 

324-if 

13. 

A of 22j-v-i% of |-f 

26. 

40,000-.002 

39. 

484-A 


241. To divide an integer or a mixed number by a mixed 
number. 

l. Divide 24 by 


Write the dividend and divisor as for division of simple 
numbers. The divisor being a mixed number, it is first 
necessary to “clear of fractions.” This is done by multi¬ 
plying the divisor by 2, the denominator of the fraction in 
the divisor. Multiplying the divisor by 2 gives 15. If 
the divisor is multiplied by 2, the dividend must also 


71)24 
2 2 

JL 

15)48 

45 

3_1 

15 ~~5 





DIVISION OF FRACTIONS 


113 


be multiplied by 2 (why?), which gives 48. The divisor and the dividend 
are now both integral, and the division is easily completed. 


2. Divide 19.44} by 4.16f. 6 

4.16f) 19.44| 

9 9 

4.66f 


Multiply both dividend and divisor by 9, the least 37 x 5.0) 175,0.00 


common multiple of the denominators of the fractions 
| and |, retaining the decimal points in the products. 
Move the points each one place to the right, and 
divide as already explained. The quotient is 4.66f. 


150 0 
25 0 0 
22 5 0 
2 5 00 
2 2 50 

2 50 

3 75 


2 

3 


242. Find the value of: 


1 . 

472£ 

+24* 

16. 978 

-851 

31. 

963f 

+ 8* 

2 . 

867 

+ 15* 

17. 125 

+1* 

32. 

248f 

+ 19* 

3. 

978§ 

v 19* 

18. 250 

-1.66f 

33. 

732| 

+ 19* 

4. 

5.26| 

+ .46| 

19. 375 

-3.331 

34. 

864| 

+ 21* 

5. 

8.49* 

+ 2.12| 

20 . 964f 

-161 

35. 

438} 

+219f 

6 . 

349| 

+ 16* 

21 . 493| 

— 42f 

36. 

644f 

+ 16* 

7. 

8749 

-j-22j 

22. 967} 

-91} 

37. 

849f 

+ 13| 

8 . 

8396 

+ 78| 

23. 83.92 

-5.125 

38. 

6741 

+ 11* 

9. 

7469| 

-=-152* 

24. 49.87} 

-2.49f 

39. 

84.96} 

+ 15* 

10 . 

9119# 

+428* 

25. 39.371 

-1.31} 

40. 

60.83} 

+ 12* 

11 . 

16.31* 

+ 1.21* 

26. 478} 

-21# 

41. 

.0482 

-K0002 

12 . 

329f 

-t-26* 

27. 674f 

-911 

42. 

59.68} 

+ .0005 

13. 

897| 

+31* 

28. 845f 

-14} 

43. 

11.25 

+ .003* 

14. 

598* 

+ 24* 

29. 7689 

— 18f 

44. 

131 

+ .008* 

15. 

939f 

+42# 

30. 598f 

-14} 

45. 

45.75 

+.001* 

Vj 

IN TUY] 

VS NEW COMP. 

AR.—8 











114 


FRACTIONS 


243. Fraction review drill. 


Add at sight, (a) vertically, (b) horizontally: 


1. 

2. 

3. 


4. 

hi 

1 5 

¥> ¥ 

24, 34 


i) it \ 

1 1 i 

1 1 

¥> ¥ 

44, 2* 


£> h ¥ 

I? h 

¥ 

34,54 


3 1 1 

Hi 4> ¥ 

i i 

4) 8 

3 1 

T) ¥ 

71, 4| 


1 _3_ I 

2 J 10) 4 

3 7 

T 1 ¥ 

1 2 
¥> ¥ 

6f, 5J 


5 7 1 

8 ) T¥) ¥ 

5 1 

8 ) ¥ 

¥> ¥ 

74, 54 


3 1 1 

4) 4) ¥ 

7 3 
¥> T 

2 1 

¥> ¥ 

84,74 


3 2 1 

4> T> ¥ 

1 7 

¥> ¥ 

1 5 

4> ¥ 

54, 34 


19 1 

4» l¥> ¥ 

State results at sight : 




5. 

6. 

7. 


8. 

2 1 _ 9 

¥ 4 — • 

4+7=44 

4+4-4 = ? 


1+7=14 

1-4=7 

1+7=4 

4+4— A=7 


1+7 = 14 

4-4=7 

4+?=t 

1-4+4 =? 


1+7 = 14 

5 1 — ? 

¥ 2 — • 

4+7=44 

1+1-4 =? 


4+7=24 

5 2 9 

¥ ¥ * 

4+?=4 

f-4+1 =7 


4+7 = 54 

5 _1 _? 

8 2 — ' t 

4+?=t 

1-1+4 =7 


4+7=34 

Multiply at sight: 




9. 

10. 

n. 


12. 

iof 12 = ? 

8X4=? 

6X34 = ? 

2 

¥ 

0f!=? 

iof 24 = ? 

9X| = ? 

8X4f = ? 

3 

T 

off = 7 

f of 16 = ? 

7Xf = ? 

9X6| = ? 

i 

of !=? 

•§• of 27 = ? 

6Xf = 7 

12X44 = ? 

i 

¥ 

of| = ? 

f of 32 = ? 

5Xf = ? 

10X61 = ? 

5 

¥ 

of| = ? 

f of 36 = ? 

4Xf = ? 

6X74 = ? 

3 

¥ 

of 4 = 7 

13. 14. 

15. 16. 

17. 18. 

19. 

20. 

5 6 

8 2| 

34 64 

9 

7 

4 2 

¥ ¥ 

1 6 

3 3 

3* 

44 
















REVIEW DRILL 

115 

Divide, stating answers at sight: 



21 . 

22 . 

23. 

24. 

y + 3 — ? 

*+3 = ? 

8—f = ? 

3H2 = ? 

t+2 = ? 

H-2 = ? 

9 -f- } = ? 

4i-r5 = ? 

I+4-? 

«■ -3 = ? 

16-^5 =? 

4^3 = ? 

*+6-7 

* +2-T 

7-5-i = ? 

3}^5 = ? 

}—4 = ? 

i +7-? 

8-j-f = ? 

81-5 = ? 

y-S"5=? 

i +4-? 

12-f-y=? 

31-4 = ? 

In the following exercises estimate results in 

advance of writ- 

ten work: 




25. Add 

26. Add 

27. Subtract 

28. Subtract 

to 

00 

tOjM 

i 

Te- 

412f 

15* 

4.032 

11.9426 

87* 

9.68} 

m 

5.9949 



16J 

7i 



Multiply: 




29. 

30. 

31. 

32. 

41| 

32} 

271 

23* 

16 

22 

16J 

22| 

33. 

34. 

35. 

36. 

28J 

33} 

17* 

271 

3 

T 

2 

¥ 

3 

¥ 

i 

4 

Divide: 




37. 

38. 

39. 

40. 

5|62| 

9|82i 

7f|423 

17i|216f 

41. 

42. 

43. 

44. 

■i-L-g- • 2 

«2J+i 

29*+* 

541 + f 


State results at sight: 

45. 46. 


450 lb. @ $2.50 per 
650 lb. @ S3.50 per 
975 lb. @ S3.75 per 
1125 lb. @ S5.25 per 


c. 

4500 

c. 

6500 

c. 

9250 

c. 

7250 


lb. @ $4.50 per M. 
lb. @ $8.50 per M. 
lb. @ $7.25 per M. 
lb. @ $5.25 per M. 


















SOME FUNDAMENTAL PRINCIPLES OF 
ARITHMETIC 


244. Every problem in arithmetic is based on one or more 
fundamental principles or laws. A knowledge of these principles 
and of their application to the solution of problems is essential to 
success in arithmetical calculations. 

Instead of stating the principles of notation, addition, subtrac¬ 
tion, etc., in connection with these topics, it has been deemed 
advisable to group them in one chapter, in order that the proper 
emphasis may be placed upon them. 

It is suggested that special attention be given to these principles, 
as frequent reference will be made to them. 

PRINCIPLES OF NOTATION AND NUMERATION 

245. l. Each removal of any significant figure* one place to the 
left toward or from the decimal point multiplies its value hy ten. 

2. Each removal of any significant figure one place to the right 
toward or from the decimal point divides its value hy ten. 

3. Each cipher annexed to an integer multiplies its value hy ten. 

4. Each cipher prefixed to a decimal divides its value hy ten. 

PRINCIPLES OF ADDITION 

5. Only like numbers or parts of like numbers can he added. 

6. The sum is composed of the same kind of units as are the addends. 

PRINCIPLES OF SUBTRACTION 

7. Only like numbers or parts of like numbers can he subtracted. 

8 . The remainder is composed of the same kind of units as are the 
minuend and subtrahend. 

9. The minuend is equal to the sum of the subtrahend and the 
remainder. 

*A significant figure is a figure that possesses a value of its own. Thus, all 
the figures, 1, 2, 3, etc., except 0, are significant figures. 0 has no value of its 
own. 


116 


FUNDAMENTAL PRINCIPLES OF ARITHMETIC 


117 


PRINCIPLES OF MULTIPLICATION 

10 . The multiplicand may be either an abstract or a concrete num¬ 
ber. 

11. The multiplier is an abstract number. 

12. The multiplicand and the product are like numbers. 

13. The multiplier and the multiplicand may be interchanged with¬ 
out affecting the value of the product. (This principle is called the 
commutative law of multiplication. 

14. The product of two factors divided by either of them will give 
the other one. (If there are more than two factors in any product , the 
product divided by any one of the factors will give a quotient equal to 
the product of all the other factors.) 

PRINCIPLES OF DIVISION 

15. The dividend is equal to the product of the divisor and the quo¬ 
tient (plus the remainder , if any). 

16. The dividend and the remainder are like numbers. 

17. Multiplying the divisor divides the quotient by the same number. 

18. Dividing the divisor multiplies the quotient by the same number. 

19. Multiplying the dividend multiplies the quotient by the same 
number. 

20. Dividing the dividend divides the quotient by the same number. 

21. Multiplying or dividing both divisor and dividend by the same 
number does not change the value of the quotient. 

Remark. Some of the above principles are already familiar to the student 
and are applied unconsciously to the solution of problems. Others (Nos. 11, 
12, 14, 15, 16, 17, 18, 19, 20, and 21) are not so familiar, and need emphasis. 
In the number of its practical applications, Principle No. 14 is the most 
important. 

APPLICATION OF THE FUNDAMENTAL PRINCIPLES OF ARITHMETIC 
TO THE SOLUTION OF PROBLEMS 

246. The solution of problems requires a knowledge of three 
things: (1) The student must know how to add, subtract, multiply, 
and divide; (2) he must know in what order these operations are 
to be performed; and (3) he must be able to make the right combi¬ 
nations of numbers in any given problem. 


118 APPLICATION OF FUNDAMENTAL PRINCIPLES 


The order of performing the operations and the proper combin¬ 
ing of numbers depend upon a knowledge of the fundamental 
principles of arithmetic. 

The following illustrations show the practical application of the 
most important principles to the solution of problems. 

Very many problems are reducible to one or more of three type 
forms. As type problems, the following may be taken as examples: 

Type l. £ of $48 = ? 

Type 2. f of what number = $45? 

Type 3. What part of $90 = $75? 

In the first example given, the operation is obviously one of mul¬ 
tiplication, and the product is $36 (Prin. 12). 

In the second example one factor and a product are given. The 
product is $45, and the given factor is f. The missing factor is 
indicated by the words “what number.” By Principle 14, if $45 
is divided by f, the other factor will be obtained. $45 -f- f = $72. 

Many problems are reducible to this type form. It is worthy of 
the careful attention of the student. 

In the third example there are also given a product, $75, and a 
factor, $90. The product divided by the given factor gives the 
other factor. $75 -s- $90 = = t* i is the missing factor. 

In the second example the missing factor was the multiplicand 
(Prin. 12). In the third example it was the multiplier (Prin. 11). 

Practice in reducing problems to their appropriate type form will 
be found helpful. Following are a few illustrations and explana¬ 
tions: 

1 . A farmer had 48 cd. of wood and sold £ of it. How many 
cords did he sell? 

First, after reading the problem, note the question, “How many cords did 
he sell?” The problem answers this question by saying that he “sold f of it.” 
What is “it”? “It” is 48 cd. Now put the number in the place of “it,” 
and you have f of 48 cd. = number of cords sold. The problem is now reduced 
to the form of the first type problem (Prin. 12). 

2 . A street is paved for a distance of 960 yd., which is £ of its 
length. What is the length of the street? 

Read the problem. Note the question to be answered. Is the street all 
paved? What part of it is paved? How many yards are paved? How do 


FUNDAMENTAL PRINCIPLES OF ARITHMETIC 119 


“t of the length of the street” and “960 yd.” compare? (They are alike.) 
Therefore write, 

* of the length of the street = 960 yd. or 

i of ? yards = 960 yd. 

960 yd. is the product obtained by multiplying some number by f. There¬ 
fore, 

960 yd. -f- t = 1200 yd., length of street. (Prin. 14, Type 2.) 

3. A man’s annual income is $2500, and his expenses for the 
year are $1500. What part of his income does he save? 

Read the problem till you know what it means. Note the question carefully. 

The question is, “What part of his income did he save?” He saved the 
difference between $2500 and $1500, which is $1000. He saved $1000. His 
income was $2500. 

Now write the question in the shortest way, using numbers instead of words 
where possible. 

? part of $2500 = $1000? (Prin. 14, Type 3.) 

Reduce each of the following problems to its proper type form: 

1. A merchant bought 480 yd. of muslin and sold f of it. How 
many yards did he sell? 

2. A boy’s total expenses for one year were $1280, of which 
amount $400 was spent for board. His board bill was what part of 
all his expenses? 

3. A manufacturing concern pays profits amounting to of its 
capital. If the profits are $15,000, what is the capital? 

4. A man sold f of his farm for $4612|. At the same rate how 
much was the whole farm worth? 

5. A dealer bought goods for $12|, and sold them for $15. The 
gain was what part of the cost? The gain was what part of the 
selling price? 

6. If a hatter buys a hat for $6, and has to sell it for $4|, he 
loses what part of the cost? of the selling price? 

7. The first of these lines is 1| 1923-—- 

in. long, the second l\ in. long. 1924- 

If the first line represents a merchant’s sales in 1923, and the 
second one, his sales in 1924, the amount of sales in 1924 was what 
part of the sales in 1923? The sales in 1924 were how much 
(fractional part) less than in 1923? The sales in 1923 were how 
much greater than fn 1924? 




120 FUNDAMENTAL PRINCIPLES OF ARITHMETIC 

PROBLEM ANALYSIS 

247. The analysis of a problem consists in stating the reason for 
each successive step in the solution. 

1. How long will it take a man earning SI5 a week to pay for a 
motor cycle costing $200, if his expenses are $7 a week? 

Analysis. If he earns $15 a week and spends $7 a week, he can save $8 a 
week. It will take as many weeks to pay for the motor cycle as $8 is con¬ 
tained times in $200, or 25 times. Therefore, it will take 25 weeks to pay for 
the motor cycle. 

2. If 3§ pounds of sugar cost 21 ff, how much will 12 pounds cost? 
Analysis. If 3£ pounds cost 21 fa 1 pound will cost as many cents as 3f is 

contained times in 21, or 6 times. If 1 pound of sugar costs 6 fa 12 pounds will 
cost 12 X 6 fa or 72 fa 

REVIEW PROBLEMS IN FRACTIONS 

Mental 

248. Find the results as indicated: 

I + i'> \ & h X J; i 1- 

2 * i + \\ i — \\ i X §; i -5- 

3 3.12.3 2- 3 v 2*3 2 

• 7 T 8) 7 8 > 7 S' 8 ) 7 * 8* 

4. 4} + 2}; 4| - 2\; 4J X 2J; 4J ,-s- 2\. 

5. 7i+3|;7i-3};7JX3|;7J + 3|. 

6. If f is subtracted from a given fraction, the remainder will 
be What is the fraction? 

7. f added to a certain fraction gives 1-^. What is the frac¬ 
tion? 

Analyze as many of the following problems as the teacher 
directs: 

8. If a man spends, on the average, $f a day for street car fare, 
in what time will he spend $100? 

9. A boy bought a bicycle for $45, and sold it for f of what he 
gave for it. How much less did he receive than he paid? 

10. Two men bought a motor boat for $450, each paying half. 
Later each sold one third of his share at cost. How much had 
each invested then, and what part of the boat did each own? 


APPLICATION OF FUNDAMENTAL PRINCIPLES 121 


11. I bought Ilf lb. of tea at $J a pound and 6 lb. of coffee at $f 
a pound. How much change should I receive from a ten-dollar 
bill? 

12. A young lady bought 7| yd. muslin at 8J </ a yard. What 
change ought she to receive from a half dollar and a quarter? 

13. A woman bought 10§ yd. of silk at $2J a yard, and 3§ yd. 
lining at $.25 a yard. She gave in payment three ten-dollar bills. 
What change should she receive? 

14. I sold a horse for $75, which was f of what it cost. How 
much did I lose? 

15. A crate containing 10 doz. oranges cost $4.50. If the 
oranges sell at the rate of 2 for 11 jzf, what is the gain? 

16. A bunch of 8 doz. bananas cost $1. If one dozen spoil and 
the rest are sold at the rate of 3 for 10 what is the gain? 

17. Two railroads carry merchandise 450 mi., for $3.60. If the 
first road carries it 200 mi., how much should each receive? 

18. Two men engage to do a piece of work for $157.50. The 
first works 18 da. and the other works 27 da. How shall the money 
be divided? 

19. 3§ doz. eggs at $.40 a dozen will pay for how many pounds 
of rice at 8f i a pound? 

20. I sold f of my apples for $90. At that rate, how much were 
they all worth? 

21. A merchant by selling cloth at 42 ^ a yard gained £ of the 
cost. Find the cost. 

22. I bought a quantity of cloth for $180. By selling it at $2 a 
yard, I gained § of the cost. How many yards were there? 

23. A buying agent charges iV of the net cost of goods as his 
commission. If his commission for one month amounts to $240, 
what is the value of the goods bought? 

24. I sold y of my land for $1500. If I sold 75 A., how much 
land have I left, and how much is it worth at the same rate? 

25. A man picked f of a bushel of pears and sold \ of them for 
$.45. At that rate, how much would a bushel cost? 

26. A man had $37|, and spent $12§. What part-of his money 
did he spend? 


122 FUNDAMENTAL PRINCIPLES OF ARITHMETIC 


27. A clerk earns $87 \ a month, and spends $30 for board and 
$20 for other expenses. What part of his money does he save? 

28. In a certain factory 56 men were employed, During a dull 
season 24 of them were laid off. What part of the men continued 
working? 

29. In an orchard there are 120 trees; 96 of them are apple trees, 
16 are pear trees, and the remainder are cherry trees. What part 
of all is each kind of trees? 

30. Three railroads carry a piece of freight 540 mi. The first 
carries it 180 mi., the second 300 mi., and the third the remainder 
of the distance. What fractional part of the distance does each 
carry it? 

PROBLEMS 

(Solve mentally when possible.) 

249. l. | of a pole 32 ft. long is decayed. How many feet are 
good? 

2. A tree 76 ft. high breaks 19 ft. from the ground. What part 
is broken off? 

3. A man has | of his money m one bank, of it in another, and 
the remainder in a third bank. If the amount in the first bank is 
$660, how much is there in each of the other banks? 

4. A man sold a horse for $150, which was f of what he paid for 
it. Find the loss. 

5. The wholesale price of Rio coffee at New York increased from 
7| i in August, one year, to 8| in August, the following year. 
What fractional part of the first price was the increase? 

6. A farmer sold f of his sheep for $360. At the same rate what 
is the value of all his sheep? What is the value of those he did 
not sell? 

7. A boy spent f of his money for a bicycle, and had $24 left. 
How much money had he at first? 

8. A man had | of his sheep in one pasture, J of them in another 
pasture, and the rest of them, which was 33 sheep, in a third pas¬ 
ture. How many sheep had he? 


APPLICATION OF FUNDAMENTAL PRINCIPLES 123 


9 . A gentleman’s estate is so divided that f of it is woodland, 
} of it is water, and the remaining 600 A. are under cultivation. 
How many acres are there in the estate? 

10. A house and lot are worth $6300. If the lot is worth -f as 
much as the house, how much is each worth? 

11. A gentleman left his son $6500, which amount was times 
as much as he left to his daughter. How much did the daughter 
receive? 

12. A owned y of a farm and B the remainder. If A had 24f A. 
more than B, how many acres had each? 

13 . A house and a barn are valued at $14,805. Find the value 
of each if the barn is worth f as much as the house. 

14 . Find the cost of each, if a house and lot together cost 
$14,000, the lot costing .4 as much as the house. 

15 . A gentleman left his estate to his three sons. The eldest 
received .40 of it, the next son .35, and the youngest, $7500. Find 
the value of the estate. 

16. The cost of furnishing two rooms is $480. If the cost of one 
is i as much as the cost of the other, what is the cost of each? 

17 . The income from a certain business for two years was 
$27,000. Find the income for each year if the second year’s 
income was .20 greater than the first. 

18. What number increased by f of itself is 651? 

19. If goods are sold for $90, and a profit of -J- of the cost is made, 
find the cost. 

20. If the profit on eggs is 3 i per dozen, and the eggs sell for 45 $, 
a dozen, what part of the cost is gained? 

21. The profit on a pair of shoes is $.75. If the profit is .20 of 
the cost, for how much per pair do the shoes sell? 

22. What number decreased by y of itself is equal to 56? 

23 . Find the cost of an article sold for $36, if \ of the cost was 
lost. 

24 . A hatter marked all his straw hats down What was the 
marked price of a hat that he now sells for $4.50? If he made a 
profit of | of the cost before marking the hat down, what part of 
the cost does he make after marking it down? 


124 FUNDAMENTAL PRINCIPLES OF ARITHMETIC 


25. If a man buys 868J yd. of cloth and sells § of it at 14J £ 
a yard, and the remainder at 12f ff a yard, how much does he 
receive for all of it? 

26. A field of 9§ A. was bought for $380. The first year’s 
wheat crop averaged 20 bu. to the acre. If wheat was worth 
90 i per bushel, the value of the crop was what part of the pur¬ 
chase price? 

27. Find the total value of the following articles: If doz. but¬ 
tons at 15 ; 2f yd. ribbon at 9| ff; 3| yd. lace at 65 jr, f yd. velvet 
at $2.90. 

28. I bought 360 sheep for $1200. I sold f of them at $4 per 
head, and J of the remainder died. At what price per head must 
the rest of them be sold to gain £ of the cost of all? 

29. A fruit dealer buys 10 doz. oranges for $2.30. If 2 doz. 
spoil, at what price per dozen must he sell the good ones to gain 
£ of the cost? 

30. From the sum of 36f-j-47£T-18f take the sum of 31§ and 
24f. 

31. A coffee importer purchases green coffee at 16£ ^ a pound. 
Freight and other expenses amount to 3| jt a pound. If, in roast¬ 
ing, the coffee loses £ of its weight, find the cost of a pound of 
roasted coffee. 

32. A young man buys 13f A. of land at $36.50 per acre. If 
he works for 27| ff an hour, 8 hr. a day, and spends $6.60 a week 
for board and other expenses, how long will it take him to pay for 
the land? 

33. If eggs are bought at 42 ^ a dozen and sold at the rate of 8 
eggs for 30 find the gain on 500 doz. eggs. 

34. In the following time sheet, 48 hours make a week’s work, 
\ extra is allowed for over time. Find weekly wages of each man: 


Names 

M. 

T. 

W. 

Th. 

F. 

S. 

Weekly 

Wage 

A. Anderson .... 

8 

9 

9 

8 

9 

10 

$36 

B. Brown. 

9 

8 

9 

8 

10 

10 

30 

C. Custer. 

10 

9 

9 

8 

8 

10 

25 

G. French. 

8 

7 

10 

9 

8 

9 

32 














APPLICATION OF FUNDAMENTAL PRINCIPLES 125 


35. Out of 900 bushels of potatoes put in storage October 15, 
45 bushels were found unsound April 1. What fractional part of 
the whole were sound? 

36. If an investment of $22,500 produces an annual income of 
$1800, how much should an investment of $25,000 produce at 
the same rate? 

37. Of a stock of cloth containing 1728 yd., 720 yd. were sold 
at one time, and 576 yd. at another time. What part of the cloth 
remained unsold? 

38. To excavate a certain basement required the removal of 
5648 cm yd. of rock and earth. If § was rock and cost $4.25 per 
cu. yd. and the remainder was earth costing 80^ per cubic yd., 
what was the cost of digging the basement? 

39. A man exchanges 8| bushels of potatoes at $1.20 per bu. 
for 24 pounds of coffee at 27§ i a pound, and 4J pounds of tea. 
How much was the tea worth per pound? 

40. A lady paid for furs $157.10; for jewelry $219.10; for shoes 
$22.80; for millinery $35; and had expended but of her money. 
How many dollars had she at first? 

41. By selling a house for $2400 I lost $600. What part of the 
cost did I lose? 

42. A quantity of wheat contains 4250 bushels. T ^- of it is 
sold at $1.25 per bushel. One half of the remainder is sold at 
$1.15, and the rest at $1.20. Find the total selling price. 

43. A merchant has three pieces of silk containing 13J yd., 
21J yd. and 16^ yd. respectively; at $1.75 a yard, how much must 
be paid for these three pieces? 

44. The last reading of my gas meter was 67,300 cu. ft.; the 
previous reading was 64,900 cu. ft. At $1.35 per thousand cu. 
ft., find the amount of my gas bill. 

45. At $64 per ton, find the cost of fertilizer for 600 young 
peach trees, allowing 1J lb. to a tree. 

46. C sells a machine for $3217.50, and thereby loses J of its 
cost. How much money did he lose, and how much did he pay 
for the machine? 

47. B sells a machine for $719.10 at a gain of i of the cost. 
How much did he gain and what was the cost price? 


126 FUNDAMENTAL PRINCIPLES OF ARITHMETIC 


48. Find the total cost of 4325 pounds of coal at $10.75 a ton; 
6284 feet of lumber at $42.50 a thousand; 3684 pounds of pork 
at $19.50 a hundred. 

49. A dealer bought 38 bags of potatoes each containing 2f 
bu. at $.87§ a bushel. He sold them at a gain of of the cost. 
Find the gain and the selling price of all. 

50. A merchant imported 1260 bags of coffee. He sold at one 
time 300 bags, and at another time re of the remainder. What 
part of the coffee remained unsold? 

51. A hatter marked his straw hats down i to close out his stock. 
His profit was then $.20 per hat, which was yo of the cost. At 
what price did he sell them before marking them down? 

The following is a form used in making an annual statement 
of receipts and disbursements by the business manager of a high 
school paper: 

Tenth Annual Financial Statement of the Business Manager of 
The High School News for the Year ended June, 1924. 



Receipts 

Disbursements 

Balai 

stce 

Sales 

Ads 

Mdse. 

Total 

Printing 

Mdse. 

Total 


Balance 















275 

90 

September 

98 

60 

15 


12 

50 

126 

10 

110 

60 



110 

60 

15 

50 

October 

102 

30 

25 

75 

60 

40 

188 

45 

112 

70 

42 

25 

154 

95 

33 

50 

November 

127 

40 

35 

50 

62 

30 

225 

20 

121 

50 

35 

20 

156 

70 

68 

50 

etc. 


















52. Prepare a form similar to the above and tabulate the re¬ 
ceipts and disbursements for the football game between the Com¬ 
merce High School and the Central Business College. There were 
sold to members of the athletic association 1528 tickets at 25^; 
to other students 732 tickets at 35 and at the gate 2406 tickets 
at 50 (jt. The expenses were for printing and advertising, $45; for 
the use of the athletic field, one third of the total receipts from 
ticket sales; miscellaneous expenses, $28.75. 

The same general form may be used for making a statement of 
receipts and disbursements in your school lunch room. 




























AVERAGE 


127 


250. l. A man sold 5 cows for $100, $127.50, $140, $165, and 
$170 respectively. Find the average selling price per cow. 

$100+$127.50-{-$140 +$165-{-$170 = $702.50, selling price of all. 

$702.50-r-5 = $140.50, average price. 

To find the average of several values, divide the sum of the 
values by the number of values. 

2. In driving from New York to Chicago an automobilist’s 
speedometer registered on successive days as follows: 215 mi., 
175 mi., 130 mi., 190 mi., 121 mi., and 217 mi. How many miles a 
day did he average? 

3. In one year a man received monthly checks as follows: 
$29.87, $98.63, $97.85, $111.54, $128.03, $101.48, $120.26, $142.38, 
$107.75, $116.11, $134.60, and $139.67. Find the average amount 
received per month. 

4. In a class of 40 students, 16 were boys and 24 were girls, 
written in the school records, for convenience, thus, 16-24. The 
daily attendance for 1 week was as follows: 15-22, 14-23, 16-21, 
13-24, 16-24. Find the average daily attendance for boys and 
for girls and the average for the class. Which group had the 
better average? 

5. During a given term of school a pupil had seven tests in 
business arithmetic, with ratings as follows: 90, 86, 81, 95, 63, 
72, 85. What was the average rating? 

6. A tire costing $28.75 was put on an automobile when the 
speedometer registered 13,496 miles. At the time of its removal, 
owing to its being worn out, the speedometer registered 18,932 
miles. Assuming that the other three tires wore equally well, 
find the average tire cost per mile of driving the car, correct to 
the nearest tenth of a mill. 

7. On the first Saturday of each month of a given fiscal year 
the wholesale weekly price of rubber per pound, at New York 
City was $1.45, $1.87, $1.92, $2.03, $1.84, $1.75, $1.76, $1.84, 
$2.06, $2.80, $2.47, $2.28, respectively. Find (1) the average 
price for the year; (2) what fractional part the average price is 
greater than the first price given. 


RATIO AND PROPORTION 

RATIO 


251. Ratio is the relation existing between two quantities of 
the same kind expressed as the quotient of the first divided by 
the second. 


252. The sign of ratio is the colon (:). 

The ratio of 8 to 4 is written 8:4. It equals f, or 8-^4, or 2. 
The ratio of 4 to 7 is written 4:7, and equals y. 


253. The terms of a ratio are the numbers compared. 

The first term is the antecedent; the second term is the con¬ 
sequent. 


254. The foregoing statements may be represented graphically, 
thus: 

^ 5 antecedent dividend numerator 

The ratio of 5 to 8 = -= —-= -=--:- 

8 consequent divisor denominator 


255. A direct ratio is the quotient of the antecedent divided by 
the consequent. 

256. An inverse ratio is the quotient of the consequent divided 
by the antecedent. 


257. Since a ratio may be expressed in the form of a common 
fraction, all the principles of fractions apply to ratio. 

258. Ratio may be simple or compound. 7 : 9 is a simple ratio; 
^ IQ is a compound ratio. 

259. Find the ratio of: 

1. 4 to 8; 5 to 6; 7 to 9; 8 to 6. 

2. 6 to 18; 8 to 32; 45 to 15; 56 to 7. 

3. 9 to 30; 30 to 9; 128 to 4; 5 to 125. 


128 





PROPORTION 


129 


s w 

4 _ 1 2 

5 “ T5* 

2 1 0 

*1:5-12:10Or,A^|=| X 


— ^ — 17T. 


Fractions having a common denominator are in the ratio of their numerators. 
Fractions not having a common denominator should be reduced to equivalent 
fractions having a common denominator, then they will be in the ratio of their 
numerators. Or the antecedent should be divided by the consequent. 


5. 8 to T to -g-J to y tO -y^-. 

6. § to -J; y^-to-j; yy to -yg-J 2 ^ to f-. 

7. 4:7 
8:12 

8 _ 8 Arrange all the antecedents as a dividend and all the 

7XZ2 21 consequents as a divisor, and apply cancellation. 

3 

8. 3:6 . 5:8 . 9:4. 7:14 

9:12' 7:14’ 12:2 J 12: 2' 


PROPORTION 

260 . Name two numbers that have the same ratio as 4 to 8; 
as 6 to 10; as 9 to 6; as 12 to 4. 

261 . Porportion is an equality of ratios. 

262 . The sign of proportion is the double colon (::). Thus, the 
expression 4:7 :: 8 :14 is a proportion, and is read 4 is to 7 as 8 is 
to 14. 

The sign of proportion is sometimes written as a sign of equality 
( = ), from which it is derived. In the proportion given, the ratio 
of 4 to 7 ($) = the ratio of 8 to 14. (t=it)- 

263 . The first and third terms of a proportion are the ante¬ 
cedents, and the second and fourth terms are the consequents. 
In the above proportion 4 and 8 are the antecedents, and 7 and 
14 are the consequents. 

264 . The first and fourth terms are the extremes; the second 
and third terms are the means. Thus, 4 and 14 are the extremes; 
7 and 8 are the means. 

VAN TUYL’S NEW COMP. AR.—9 



130 


RATIO AND PROPORTION 


265 . The solution of problems in proportion depends upon the 
following principles: 

1. The product of the means equals the product of the extremes. 

2. The product of the means divided by either extreme gives the 
other extreme. 

3. The product of the extremes divided by either mean gives the 
other mean. 

266. Find the missing term in each of the following: 

1. 14 : 25 : : 21 : ? 

25X21 _To find the missing extreme, divide the product of the 

14 2 ‘ means by the given extreme, 14. The other extreme is 37£. 

2. ? : 54 :: 28 :42. 5. 75 : ? : : 25 :40. 

3. 32 : 50 :: ? : 25. 6. 16J : 24 f :: 30 : ? 

4. 55 : 60 :: 11 : ? 7. 33£ : 50 :: ? : 200. 

267 . Proportion may be simple, compound, or partitive. 

SIMPLE PROPORTION 

268 . Simple proportion is an equality of simple ratios. All the 
proportions given above are simple. 

Cause and Effect 

269 . The simplest method of making a proportion statement 
from a given problem is by the method of “Cause and Effect.” 
In every problem there are certain “causes” the operation of which 
results in certain “effects.” Every effect, or result, has a cause, and 
every cause produces an effect. In any given problem, the effects 
are in the same ratio as the causes. 

270 . If 11 bbl. of flour cost $60.50, how much will 21 bbl. cost 
at the same rate? 

1st 2d 1st 2d 

cause cause effect effect 

11 bbl. : 21 bbl. : : $60.50: $? 

5.50 

21 XS 60 .g 0 = $1155O 





PROPORTION 


131 


The 11 bbl. is the 1st cause, the effect of which is an expenditure of $60.50. 
The 2d cause is the 21 bbl., the effect, or cost, of which is to be found. Divid¬ 
ing the product of the means by the given extreme, the cost of 21 bbl. is found 
to be $115.50. 

Note. Observe that the two ratios are each composed of like quantities. 
If this fact is kept in mind, it will aid the student in making statements of 
proportion. 

PROBLEMS 

271 . l. If 17 A. of land produce 442 bu. of grain, how many 
bushels will 51 A produce? 

2. 18 horses eat 23f bu. of grain a week. How much will 27 
horses eat in 8 wk.? 

3. If 24 yd. of cloth cost $2.64, how much will 33 yd. of the 
same kind of cloth cost? 

4. At 45 0 a dozen how much will 16 oranges cost? 

5. The shadow of a post 9 ft. high is 10 ft. long. How high is a 
tree that casts a shadow 90 ft. long at the same time? 

6. For how long a time should $500 be loaned to balance a loan 
of $800 for 10 mo.? 

7. The capacity of a shoe factory is 2580 pairs per week of 6 
da. How long will it take to fill an order for 34,400 pairs? 

8. If 12 men can do a piece of work in 9 da., how long will it 
take 18 men to do it? 

9. It is estimated that 45 men can do a piece of work in 12 da. 
How many additional men should be employed to perform the 
work in 9 da.? 


COMPOUND PROPORTION 

272. Compound proportion is an equality of compound ratios. 
Thus, 4 :8 

7 :21 : : 15 :90 

9 :18 4:8 is a compound proportion. 

273. The principles underlying solutions in compound propor¬ 
tion are the same as those applied in simple proportion, 


132 


RATIO AND PROPORTION 


274 . If an army of 2500 men in 15 da. requires 46,875 lb. of 
food, how many pounds of provisions will 3500 men need in 25 da.? 

1st cause 2d cause 1st effect 2d effect 

2500 men: 3500 men :: 46875 lb.: ? lb. 

15 da.: 25 da. 


3125 

3500X20X40075 lb. 


= 109,375 lb. 


2500XZ5 

The 1st cause is the 2500 men requiring food for 15 da., as a result of which 
46,875 lb. were required. The 2d cause is similar to the 1st cause, but is 
greater, that is, 3500 men for 25 da. The second effect is to be determined. 
Dividing the product of the means by the giveta extreme gives the required 
amount of food as 109,375 lb. 

PROBLEMS 


275 . l. 75 sheets of paper 36"X54" weigh 24 lb. How much 
do 5000 sheets 17"X22" of the same grade of paper weigh? 

2. If 3 doz. sheets of drawing paper 24"X36" weigh 2 lb., find 
the weight of 6 reams (500 sheets = 1 ream) 8"X12" of the same 
grade of paper. 

PARTITIVE PROPORTION 


276. l. Three partners, A, B, and C, agree that the profits of 
their business shall be divided into 9 equal parts or shares, of 
which A is to have 4; B, 3; and C, 2 shares. If the profits amount 
to $6300, how much does each partner receive? 

A has f of $6300, or $2800. Since there are 9 P arts or shares > and A 
B has|of $6300, or $2100. has 4 4 f of t * hem ’]? has J °/ a11 the shares - 

„ a ’ or i of the profits. For the same reason 

C has f of $6300, or $1400. B has and C; ot the profits . 

In the above problem, the profits are divided among A, B, and C in the 
proportion of 4, 3, and 2. 


The process of dividing a number into parts proportional to 
other given numbers may be called partitive proportion. 

2. Two men do a piece of work for $6. The first man works \ 
of a day, and the second f of a day. How shall they divide the $6 ? 

Only similar fractions can be compared. | = f, f = f. Similar fractions 
are in proportion to their numerators. Hence, \ and f are in proportion to 2 
and 3. That is, the $6 is to be divided into parts proportional to 2 and 3. 


2 parts + 3 parts = 5 parts. 

The first man has 2 parts, or § of $6, or $2.40. 
The second man has 3 parts, or * of $6, or $3.60. 



PROPORTION 


133 


277 . 1 . A worked 3 da. and B 4 da. for $28. How much should 
each receive? 

2. Two men rent a pasture for $50 A puts in 4 horses, and B 
6 horses. How much should each pay? 

3. Three men performed a piece of work for $432. The first 
man worked 24 da., the second 40 da., and the third 32 da. How 
much should each receive? 

4. Divide $540 among three families so that the first shall 
receive $2 and the second $3, as often as the third receives $4 

5. Four partners’ investment is in proportion to 6, 7, 8, and 9. 
They gain in one year $10,800. How much is each partner’s share? 

6 . Divide 25 into parts proportional to \ and 

7. Divide 35 into parts proportional to i and 

8. The sides of a triangle are in proportion to 4. 5, and 6. Its 
perimeter is 360 ft. Find the sides. 

9. Four men and eight boys earn $440. If one man earns as 
much as two boys, how much does each earn? 

10. Two building lots cost $14,000. One cost § more than the 
other. How much did each cost? 

11. A field rents for $36. B puts in 12 sheep for 15 wk. and C 
20 sheep for 18 wk. How much should each pay? 

12. Divide 180 into parts proportional to 3, lj, f, and f. 

13. Three partners agree to divide their profits in proportion to 
§, 1J, and 2. If their profits are $7050, what is the share of each? 

14. An estate of $50,000 is divided among four sons in propor¬ 
tion to their ages, which are 20 yr., 17 yr., 14 yr., and 11? yr., 
respectively. Find the share of each. 

15. A and B are partners. A invests $5000 for 8 mo. and B, 
$7500 for 6 mo. Their gains amount to $3400. Find the share 
of each. 

16. C, D, and E are partners. C invests $8000 for a year. Din- 
vests $10,000 for 10 mo., and E invests $6000 for 8 mo. Divide 
a profit of $9760 among them. 


EXAMINATIONS 

SPEED TEST 

278 . Minimum time, thirty minutes; maximum time, one hour. 
Suggestion for marking: Allow 100 credits if test is completed cor¬ 
rectly in minimum time. Deduct one credit for each minute re¬ 
quired beyond minimum time. 

2. From Take Difference 


l. Add the following: 


241693798 

$693.05 

473185459 

7.847 

391533768 

78.98 

427936857 

1489.86 

819348673 

.75 

473925165 

974.98 

274639827 

6. 

315987352 

87.87 

675431298 

9748.098 

897316984 

7.96 

931258369 

75.899 

687316984 

17. 


752361 582834 

1021314 987476 

7934.19 986.74 

784 95.9876 

3. Find the total value of the fol¬ 
lowing: 222 1 yd. @ 3 2 $£ 

166 2 yd. @ 4 1 
265 1 yd. @7H 
163 2 yd. @ 9 2 i 
2543 yd. 

212 yd. @ 4 3 £ 

5. Find the total value 
of the following: 

3290 lb. @ $2.75 per 
hundredweight. 

4332 lb. @ 2.75 per 

hundredweight. 

3570 lb. @ 1.87J per 
hundredweight 
5792 lb. @ 22.50 per 

1000 lb. 

2396 lb. @ 16.50 per 

1000 lb. 

32690 lb. @ 12.80 per 
1000 lb. 


4. Complete the following bill: 


Chicago, III., July 1, 1924. 

H. P. Jones 

Bought of 

D. C. Roy. 

3 bbl. Sugar, 640# @ 6jj£ 

5 chests Tea, 680# @ 75 £ 

3 chests Tea, 498# @ 66 

6 bbl. Flour, 1176# @ 4 i 

Total. 




134 












SPEED TEST 


135 


6. Add 873 to 1,301 continuously till the sum equals or exceeds 
10,000. (Show all the work.) 

7. Divide 23,236 by 2237 by continued subtraction. (Show all 
the work.) 

8. Reduce to equivalent fractions having the least common 
denominator: 

12 5 7 /Vv 9 £ 5 3 

\ a ) 2> 3) T2) 8- \C) T~6) 8) 2~4> 3 2‘ 

(b) .75, f, -J, t 5 q. ( d ) .80, §, ^g-, .3. 

9. Reduce to common fractions in their lowest terms: 

.065 .0625 .225 .00125 

10. Perform the operation indicated: 

15J X 17J 426f -r- 2l\ 48.72 X .00125 54.7325 .001 

SPEED TEST 

279. 1. Add horizontally and vertically. Check by adding the 


totals: 

$36,968 $ 7,634 $14,262 $- 

24,635 28,329 37,257 - 

9,564 82,378 6,295 - 

49,877 69,764 34,472 - 

Grand total.$- 

2. Find the gain on each of the following items and the total 
gain: 

Selling Price Cost Gain 

140,000.00 $37,624.37 $- 

9,437.83 5,819.24 - 

4,341.85 2,838.96 - 

57,640.33 48,924.29 - 

89,754.00 68,947.25 - 

Total.. $- 


Perform the following indicated operations: 


3. 425 -T- £ = 

234 + .33| = 
.25 X f = 
9i - 4 A = 
i+i+i= 


4. 9 X 12 X 14 42 = 

95 X 8 X 3 -5- 19 = 
52 X 8 X 7 -t- 91 — 
20 X 21 X 22 -j- 154 = 
(48 X 36 X 24) (16 X 12) = 









136 


EXAMINATIONS 


5. Find the value of each of the following: 

3 _ 1 _ 5 _ 1 13 

-2. X 12J = 7X }%xn = 

4 ^ 

6. Reduce each of the following groups to equivalent fractions 

having the least common denominators: 


9 13 7 1 1 _ 

W 27 T¥ T2 ¥¥ — 

3 4 11 9_ _ 

\P) ¥ T XT TO “ 

7. Multiply and prove: 
76897 X 328 
45723 X 126 

9. Find the total cost of the 
following: 


250 

bu. 

@ 

$ .75 

340 

bu. 

@ 

.80 

1360 

bu. 

@ 

1.12} 

1840 

bu. 

% 

1.25 

1480 

bu. 

@ 

.62} 



Total . 


(c) 

4 

TIT 

5 

¥T 

4 

2T 

2 _ 

¥3 — 

(d) 

5 

T2 

TT 

1 l 

TS 

if ~ 

8. 

Divide and 

prove: 


6806} -r- 412} 

10 . Find the total interest at 
6% on the following: 

$500.00 for 90 da. = $- 

750.00 for 80 da. =- 

1000.00 for 30 da. =- 

1200.00 for 45 da. =- 

895.50 for 20 da. = - 

Total . . $- 


WRITTEN TEST 


280 . 1 . A company employs 17 laborers at $3.50 per day; 28 
mechanics at $6.60 per day; 9 teamsters at $4.50 per day; 2 book¬ 
keepers at $25 a week; and a superintendent at $3120 a year. 
What is the weekly pay roll of the company? 

2. A man sold a farm of 248} A. at $45} per acre, and with the 
proceeds bought another farm of 125 A. What price per acre did 
he pay for the new farm? 

3. A person owned f of a mine and sold } of his share for 
$1710. What was the value of the mine? 

4. Find the net cost of the following bill of hardware: 12 kegs 
cut nails, 8d., 1200 lb. @ 5 i per pound; 5 doz. handsaws, 26 in., 
@ $25 per dozen; 4 doz. carpet stretchers at $5.50 per dozen; 20 
doz. garden rakes @ $6.25 per dozen; and 12 doz. try squares @ 
$7.50 per dozen. Less 20% on the entire bill. 





WRITTEN TEST 


137 


5. Find the total cost of the following: 8 pieces broadcloth, 56 1 , 
58 3 , 57, 59 2 , 55 1 , 60 1 , 61 2 , 58 3 yd. at $6.25; 5 pieces corduroy, 47 2 , 
44 2 , 43 3 , 46 2 , 45 1 yd. at $1.75; and 7 pieces storm serge, 43 1 , 47 2 , 
48 3 , 44, 45 2 , 48 2 , 47 3 yd. at $2.25. 

6. A man had on hand in the morning cash in the safe amount¬ 
ing to $206.20, and in the bank $1379.30. During the day he 
received cash in currency amounting to $909.24 and checks amount¬ 
ing to $489.36. Cash deposited in the bank, $604.37. Checks 
drawn on the bank account, $3.97, $47.86, $396.25, $49.83, 
$246.97. Cash paid out in bills and coin, $49.86. Show the con¬ 
dition of the cash and bank accounts in the evening. 

7. Find the total cost of 64 bags of sugar, each containing 12^ 
lb. at 131 k a pound; 19 bbl. of flour at $13.50 a barrel; 1740 eggs 
at 47 0 a dozen; 375 pineapples at $12.75 per hundred; 42 qt. of 
walnuts at 14§ i a quart. 

8. A stenographer gets $116f per month. How long will it 
take him to pay for a house and lot costing $2300, if his yearly 
expenses average $1000? Reduce time to years, months, and days. 

9. Find the cost of the following: 348 eggs at 43^ per dozen; 
643 lb. sugar at 13§^ per pound; and 7750 lb. coal at $9.75 per 
ton. 

10 . In a hospital having 1048 inmates there were consumed the 
following articles: Meat, fresh, 287,138 lb., average price, $.146; 
meat, smoked, 7614 lb., average price $.23; meat, canned, 7239 
lb., average price $.126; poultry, 4128 lb., average price, $.234. 
Find the total cost and the cost per capita. 


DENOMINATE NUMBERS 

281. A denominate number is a concrete number whose unit of 
value has been fixed by law or custom. $5; 2 qt.; 3 rd.; are de¬ 
nominate numbers. 

282. A simple denominate number is one whose value is ex¬ 
pressed in one denomination. 4 qt.; 7 in.; 8 gal.; are simple 
denominate numbers. 

283. A compound denominate number is one whose value is 
expressed in two or more denominations. 5 sq. yd., 2 sq. ft., 18 
sq. in.; 8 bu., 2 pk., 6 qt., are compound denominate numbers. 

284. A measure is a standard unit used to determine quantity. 

285. Quantity is anything that can be measured, as length, 
area, volume, capacity, time, value, etc. 

286. A standard unit of measure is a unit established by law or 
custom, by which other units are to be adjusted. 

The gallon of 231 cu. in. is the established unit of measure for liquids; from 
it the other units, gill, pint, quart, and barrel are derived. 

The yard is the standard unit of measure for length; the inch, foot, rod, and 
mile are derived from the yard, etc. 

287. A quantity is measured by determining how many times 
it contains its appropriate unit of measure. 

Linear Measure 

288. Linear measure is used for measuring distance. 

Table 

12 inches (in.) = 1 foot (ft.) 

3 feet = 1 yard (yd.) 

5§ yards = 1 rod (rd.) 

320 rods = 1 mile (mi.) 

1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63,360 in. 

The unit of length is the yard. 


138 


APPROXIMATE MEASURES 


139 


Other Linear Measures 

1 size, | in. Used by shoemakers. 

1 hand, 4 in. Used in measuring the height of horses. 

1 fathom, 6 ft. Used in measuring depths at sea. 

1 knot, nautical or geographical mile — 1.152f mi. or 6086 ft. 

The knot is used in measuring distances at. sea. It is equivalent to 1 min. of 
longitude at the equator. 

Square Measure 

289. Square measure is used to measure the areas of surfaces. 

Table 

144 square inches (sq. in.) = 1 square foot (sq. ft.) 

9 square feet = 1 square yard (sq. yd.) 

30* square yards = 1 square rod (sq. rd.) 

160 square rods = 1 acre (A.) 

640 acres = 1 square mile (sq. mi.) 

1 sq. mi. = 640 A. = 102,400 sq. rd. = 3,097,600 sq. yd. = 27,878,400 sq. ft. 
= 4,014,489,600 sq. in. 

A square is 100 sq. ft. 

In measuring land, other than city lots, the acre is the unit. 

In measuring other surfaces, the square yard is the unit (except in roofing, 
where the square is used). 

Cubic Measure 

290. Cubic measure is used to measure the volume of solids 
and the contents or capacity of hollow bodies. 

Table 

1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 

27 cubic feet = 1 cubic yard (cu. yd.). 

24* cubic feet = 1 perch (P.). 

128 cubic feet = 1 cord (cd.). 

1 cubic yard (of earth) = 1 load. 

A perch of stone or masonry is 16£ ft. (1 rd.) long, 1§ ft. wide, and 1 ft. high. 
A cord of wood is a pile 8 ft. long, 4 ft. wide, and 4 ft. high. 

A cubic foot of water weighs 62£ lb. (Avoir.). 

APPROXIMATE MEASURES 

A ton of timothy hay in a well-settled mow is about 450 cu. ft. 
A ton of clover hay in a well-settled mow is about 550 cu. ft. 

A ton of stove coal is about 34J cu. ft. 


140 


DENOMINATE NUMBERS 


Notes. Tables for surveyors’ linear and square measure are omitted be¬ 
cause they are going out of use. Surveyors now use, instead of the Gunter’s 
chain, a steel tape 100 ft. long, divided into feet and tenths of a foot (some¬ 
times into inches). Measurements are more nearly accurate by the tape 
because there are no joints to wear as in the chain. Areas are measured by 
finding the number of square feet and dividing by 43,560, the number of square 
feet in an acre. 


MEASURES OF CAPACITY 

Liquid Measure 

291. Liquid measure is used for measuring liquids, and in esti¬ 
mating the capacity of cisterns, tanks, reservoirs, etc. 

Table 

4 gills (gi.) = 1 pint (pt.) 

2 pints = 1 quart (qt.) 

4 quarts = 1 gallon (gal.) 

1 gal. = 4 qt. =8 pt. = 32 gi. 

Technically, the barrel contains 31£ gal. In practice, barrels are of various 
sizes, as are also hogsheads, pipes, butts, etc. The capacity of each is marked 
upon it. 

The unit of liquid measure is the wine gallon of 231 cu. in. A gallon of water 
weighs about 8f lb. (Avoir.). 

The imperial gallon of England contains 277.274 cu. in. and is equal, very 
nearly, to 1^ wine gallons. It contains 10 lb. of water. 


Apothecaries’ Fluid Measure 

292. Apothecaries’ fluid measure is used by druggists in prescrib¬ 
ing and compounding liquid medicines. 

Table 

60 minims (m.) = 1 fluid drachm (f 5) 

8 fluid drachms = 1 fluid ounce (f 5) 

16 fluid ounces = 1 pint (O) 

8 pints = 1 gallon (Cong.) 

1 Cong. = 8 O = 128 f 5 = 1024 f 5 = 61,440 m. 

The gallon of this measure is the same as the wine gallon. 


WEIGHT 


141 


Dry Measure 

293. Dry measure is used for measuring grain, fruit, vegetables, 
etc. 

Table 

2 pints (pt.) = 1 quart (qt.) 

8 quarts = 1 peck (pk.) 

4 pecks = 1 bushel (bu.) 

1 bu. = 4 pk. = 32 qt. = 64 pt. 

The unit of dry measure is the Winchester bushel, which contains 2150.42 
cu. in. It is a hollow cylinder 18£ in. in diameter and 8 in. deep. 

The Winchester bushel is used in measuring grain, sand, etc. 

The heaped bushel of 2747.71 cu. in. is used for measuring fruits, vegetables, 
and other coarse articles. 

The imperial bushel of England contains 2218.192 cu. in. It is exactly 8 
times the imperial gallon. 


Liquid and Dry Measures Compared 

Gallon Quart 

Liquid . . . 231 cu. in. 57f cu. in. 

Dry .... 268* cu. in. (I peck) 67|cu. in. 


Pint 

281 cu. in. 
33f cu. in. 


Weight 

294. Weight is the measure of the earth’s attraction of matter 
on its surface. 

295. There are four kinds of weight in use in the United States: 
Troy, Apothecaries’, Avoirdupois, and Metric.* 

296. The Troy pound is the standard unit of weight in the 
United States. 

Troy Weight 

297. Troy weight is used in weighing gold, silver, diamonds, 
and other precious minerals. It is used by the government in 
weighing coins at the mint, by the jewelry trade and manufactures, 
and by importers and exporters of gold and silver. In the jewelry 
trade and manufacturing industries where gold, silver, and pla¬ 
tinum are employed, a thorough knowledge of the Troy units 
and their comparisons is important. 

♦Metric tables are given on pages 148-151. 


142 


DENOMINATE NUMBERS 


Table 

24 grains (gr.) = 1 pennyweight (pwt.) 

20 pennyweights = 1 ounce (oz.) 

12 ounces = 1 pound (lb.) 

1 lb. = 12 oz. = 240 pwt. = 5760 gr. 

The term “carat” (or karat) has two meanings. In weighing 
diamonds, it is a denomination of weight, and is equal to 3.168 
gr. Its second use is to denote the fineness of gold, and means -Jt 
part. Gold marked 18K (18 carats) is |-f, by weight, pure gold 
and gr alloy. 

Apothecaries’ Weight 

298. Apothecaries’ weight is used by druggists and physicians 
in compounding and prescribing medicines. 


Table 

20 grains (gr.) = 1 scruple (sc. or 9) 

3 scruples = 1 dram (dr. or 5) 

8 drams = 1 ounce (oz. or 5) 

12 ounces = 1 pound (lb. or lb) 

1 lb. = 12 5 = 96 5 = 288 ©= 5760 gr. 

Drugs and chemicals are bought and sold wholesale by avoirdu¬ 
pois weight. 

Avoirdupois Weight 

299. Avoirdupois weight is used in weighing all sorts of coarse, 
heavy articles. 

Table 

16 ounces (oz.) = 1 pound (lb.) 

100 pounds = 1 hundredweight (cwt.) 

20 hundredweight = 1 ton (T.) 

1 T. = 20 cwt. = 2000 lb. = 32,000 oz. 


Long Ton 

300. The long ton (or gross ton) contains 2240 lb. It is used in 
the United States customhouse in determining the duty on mer¬ 
chandise taxed by the ton. Coal and iron are sold wholesale at 
the mines by the long ton. 


COMPARISON OF WEIGHTS 


143 


Table 

16 ounces (oz.) = 1 pound (lb.) 

28 pounds = 1 quarter (qr.) 

4 quarters = 1 hundredweight (cwt.) 

20 hundredweight = 1 ton (T.) 

1 T. = 20 cwt. = 80 qr. = 2240 lb. = 35,840 oz. 


Comparison of Weights 



Pound 

Ounce 

Grain 

Troy 

5760 gr. 

480 gr. 

1 gr. 

Apothecaries’ 

5760 gr. 

480 gr. 

1 gr. 

Avoirdupois 

7000 gr. 

437^ gr. 

1 gr. 


Note that the pound and the ounce of Troy and apothecaries’ 
weights are alike, and that the grain is the same in all three kinds 
of weight. 


301. Table showing Weights in Pounds per Bushel 

LEGALLY ESTABLISHED FOR CERTAIN PRODUCTS BY THE SEVERAL 

States and (for Customs Purposes) by Congress 



U. S. 

States 

Exceptions 

Barley .... 

48 

48 

Ala., Ga., Ky., Pa., 47; Ariz., 45; Cal., 50. 

Beans .... 

60 

60 

Ariz., 55; N. H., Vt., 62. 

Buckwheat . . 

48 

52 

Cal., 40; Conn., Me., Mass., Mich., Miss., 
N. Y., Pa., R. I., Vt., 48; Idaho, N. Dak., 
Okla., Ore., S. Dak., Tex., Wash., 42; 
Ind., Kan., Minn., N. J., N. C., Ohio, 
Tenn., Wis., 50. 

Clover seed . . 

60 

60 

N. J., 64. 

Corn (in ear) . 

70 

70 

Miss., 72; Ohio, 68. 

Corn (shelled). 

56 

56 

Mass., 50. 

Corn meal . . 

48 

50 

Ark., Fla., Ga., Ill., Miss., S. C., 48. 

Oats. 

32 

32 

N. J., Va., 30. 

Onions .... 

57 

57 

Conn., Me., Mass., Minn., N. Dak., Okla., 
S. Dak., Vt., 52; Fla., Tenn., 56; Ind., 
48; Mich., 54; Ohio, 55; Pa., R. I., 50. 

Peas. 

60 

60 


Potatoes . . . 

60 

60 

Md., Pa., 56. 

Rye. 

56 

56 

Cal., 54; Me., 50. 

Timothy seed . 

45 

45 

Ark., 60; Okla., S. Dak., 42. 

Wheat .... 

60 

60 
















144 


DENOMINATE NUMBERS 


Other Common Measures 


Pounds 

Beef, Barrel.200 

Butter, Firkin.56 

Flour, Barrel.196 

Nails, Keg.100 

Pork, Barrel.200 

Salt, Barrel.280 


Circular or Angular Measure 

302. Circular or angular measure is used in measuring angles or 
arcs of circles as applied to surveying, civil engineering, astronom¬ 
ical calculations, latitude, longitude, etc. 


Table 

' 60 seconds (") = 1 minute (') 

60 minutes = 1 degree (°) 

360 degrees = 1 circle (cir.) 

1 cir. = 360° = 21,600' = 1,296,000". 

In astronomical calculations the circle is sometimes 
divided into signs, sextants, and quadrants. 

In a circle are: 12 signs of 30° each. 

6 sextants of 60 3 each. 

4 quadrants of 90° each. 

303. The unit of circular measure is the degree, and is equal to 
of the circumference of a circle. 

304. On the earth’s surface at the equator 1° of distance is 
equal to 69J statute miles, or 60 geographical miles or knots. 
Hence, 1 min. of distance equals 1 knot. 

305. In measuring circles the length of a degree is dependent 
upon the size of the circle, but in measuring angles, the degree has 
a fixed value. 

Time 

306. Time is the measure of duration. 



Table 


60 seconds (sec.) = 1 minute (min. 
60 minutes = 1 hour (hr.) 

24 hours = 1 day (da.) 

7 days = 1 week (wk.) 

30 days = 1 month (mo.) 


52 weeks = 1 year (yr.) 

12 months = 1 year 

365 days = 1 common year 

366 days = 1 leap year 
100 years = 1 century 










TIME 


145 


307, The months of the year. 

January (Jan.) 31 days 

February (Feb.) 28-9 days 
March (Mar.) 31 days 

April (Apr.) 30 days 

May 31 days 

June 30 days 


July 31 days 

August (Aug.) 31 days 
September (Sept.) 30 days 
October (Oct.) 31 days 
November (Nov.) 30 days 
December (Dec.) 31 days 


The length of the year is determined by the time required for the earth to 
make one complete revolution around the sun. The exact time is 365 da. 5 hr., 
48 min. 46 sec. (very nearly 365i- da.). 

It is because of this extra \ da. each year that we have a leap year once in 
4 yr. But by allowing | of a day each year in making a leap year each 4 yr., 
a little too much time is allowed, as the true year lacks a little of being 365£ da. 
To correct the error, all centennial years (1800, 1900, 2100) not divisible by 
400 are not leap years. All other years divisible by 4 are leap years. 

The extra day of leap year is added to the month of February. 


Standard Time 

308. Since the earth makes a complete rotation of 360° in 24 
hr., any point on the earth’s surface passes through 15° of space in 
1 hr. The earth rotates from west to east, and makes the sun 
appear to “rise” in the east and “move” toward the west. From 
New York to San Francisco is a distance of about 50°, or a dif¬ 
ference in time of about 3J hr. Hence the sun rises in New York 
3§ hr. before it does in San Francisco. A person traveling from 
New York to San Francisco would find that his watch was too 
fast. It has seemed to gain an hour for every 15° of distance. 

To remedy this condition, the principal railroads of the United States in 
1883 agreed upon a system of reckoning time since known as “Standard Time.’' 
The territory of the United States was divided into four north and south belts. 
These belts lie approximately half and half on each side of the 75th, the 90th, 
the 105th, and the 120th meridians of longitude, respectively. It was agreed 
that the time for all places approximately 7§° either east or west of any one of 
these meridians should have the same time as that of the chief meridian for 
that belt, and that there should be a difference of 1 hr. in the time between 
any place in one belt and any place in an adjoining belt. 

The time for all places in the belt having the meridian of 75° longitude west 
from Greenwich as the chief meridian is called Eastern Time; for all places in 
the belt having the meridian of 90° as the chief meridian, as Central Time, 
for the belt having the meridian of 105° as the chief meridian, as Mountain 
Time; and for the belt in which the meridian of 120° is the chief meridian, as 
Western or Pacific Time. 

VAN TUYL’S NEW COMP. AR—10 


146 


DENOMINATE NUMBERS 


The dividing line between any two adjoining time belts is not a straight 
north and south line, but is very irregular, so that the change in the time can 
be made at the points most convenient to the different railroads. 

What is the difference in time between Boston and Chicago? 
Boston and Denver? New York and San Francisco? Charleston 
and Santa Barbara? Cincinnati and Denver? 



Map op Standard Time Belts 


English Money 

309. English money is the legal currency of Great Britain. 
There are gold, silver, and copper coins, and bills. 


Table 

4 farthings (far.) = penny (d.) 

12 pence = 1 shilling (s.) 

20 shillings = 1 pound, ot sovereign (£) 

The unit of English monetary value is the sovereign. It contains 113.0016 
gr. of pure gold. Its value in United States money is, therefore, as many 




MISCELLANEOUS MEASURES 


147 


dollars as 23.22 gr. is contained times in 113.0016 gr., or $4.8665. (Called par 
of exchange or par value. See page 397.) 

The gold coins are the sovereign, and the half sovereign. They are 22K 
(or H) fine. 

The silver coins are crown (5s.), half crown, florin (2s.), sixpence, and three¬ 
pence. They are .925 fine. 

The copper coins are the penny, halfpenny, and the farthing. 

French Money 

310. French money is the legal currency of France. It is a 
decimal currency. 

Table 

10 millimes (m.) = 1 centime (c.) 

10 centimes = 1 decime (dc.) 

10 decimes = 1 franc (F.) 

The unit of French money is the franc, and is equal to $.193 of United States 
money. The franc contains 4.4803 gr. pure gold. 

Gold and silver coins of France are ^ fine. 

Gold coins are 5, 10, 50, and 100 franc pieces. 

Silver coins are 25 and 50 centime pieces, and the 1, 2, and 5 franc pieces. 
Bronze coins are 1, 2, 5, and 10 centimes pieces. 

F. 5.27 is read “5 francs 27 centimes.” 

German Money 

311. German money is the legal currency of Germany. 

Table 

100 pfennig (pf.) = 1 mark (M.) 

The unit is the mark. The value of the gold mark in United States money 
is $.238. 

There are gold, silver, nickel, and copper coins. The gold and silver coins 
are fine. The mark contains 5.5313 gr. of pure gold. 

The gold coins are 5, 10, and 20 mark pieces. 

The silver coins are 20 and 50 pfennig pieces and 1 and 2 mark pieces. 
The nickel coins are 5 and 10 pfennig pieces. 

The copper coins are 1 and 2 pfennig pieces. 

Miscellaneous Measures 

312. Paper. 

Table 

24 sheets = 1 quire (qr.) 

20 quires = 1 ream (rm.) 

2 reams = 1 bundle (bdl.) 

5 bundles = 1 bale (bl.) 


148 


DENOMINATE NUMBERS 


313. Counting. 

Table 

20 units = 1 score 
12 units — 1 dozen (doz.) 

12 dozen = 1 gross (gro.) 

12 gross = 1 great gross (gr. gro.) 

THE METRIC SYSTEM 

314. The metric system is a decimal system of weights and 
measures. Nearly all the civilized nations of the world except the 
United States and England use the system. 

315. The metric system was legalized in the United States in 
1866, but has not been generally adopted except in scientific work. 

316. The fundamental unit of the system is the meter, because 
every other unit of measure or weight is based on it. 

317. The length of the meter was determined by taking one 
ten-millionth of the distance from the equator to the pole. Its 
length is 39.37 in. 

318. The advantages of the metric system are: 

(1) “The decimal relation between the units; 

(2) “The extremely simple relation of the units of length, area, volume, and 

weight to one another; 

(3) “The uniform and self-defining names of units.” 

319. The primary units are the following: 

For length, meter = 39.37 inches. 

For capacity, liter = .908 dry quarts; 1.0567 liquid quarts. 

For weight, gram = 15.432 grains. 

320. There are three Latin, and four Greek, prefixes, as follows: 
The Latin prefixes are: 

milli = (millimeter = meter) 
centi = t £o (centimeter = meter) 
deci = tV (decimeter = T V meter) 

The Greek prefixes are: 

deca =10 (decagram = 10 grams) 

hecto = 100 (hectogram = 100 grams) 

kilo = 1000 (kilogram = 1000 grams) 

myria = 10000 (myriagram = i0000 grams) 


THE METRIC SYSTEM 


149 


321. The United States government requires the use of the 
metric system of measures in all medical work of the navy and 
the war departments, and in the public health and marine hospital 
service. Its use is obligatory in Porto Rico. 

322. For postal purposes “fifteen grams .. .shall be the equiv¬ 
alent .. of one half ounce avoirdupois, and so on in progression.” 

At the mint 12| g. is the weight of a half-dollar. The quarter- 
dollar and the dime in proportion (See Weights of Coins, page 11.) 

Linear Measure 

323. The unit of linear measure is the meter. 


Table 


10 millimeters (mm.) 
10 centimeters 
10 decimeters 
10 meters 
10 decameters 
10 hectometers 
10 kilometers 


1 centimeter (cm.) 

1 decimeter (dm.) 

1 meter (m.) 

1 decameter (Dm.) 

1 hectometer (Hm.) 
1 kilometer (Km.) 

1 myriameter (Mm.) 


Notes. (1) The most commonly used denominations in the above and in 
the following tables are indicated by the heavy-faced type. 

(2) It should be observed that the abbreviations of the Latin prefixes begin 
with small letters, and those of the Greek prefixes begin with capital letters. 


Square Measure 


324. The unit of square measure is the square meter for small 
areas, and the are of 100 sq. m. for land areas. 


Table 


100 square millimeters (sq. mm.) 

100 square centimeters 

100 square decimeters 

100 square meters 

100 square decameters 

100 square hectometers 


= 1 square centimeter (sq. cm.) 
= 1 square decimeter (sq. dm.) 

= 1 square meter (sq. m.) 

= 1 square decameter (sq. Dm.) 
= 1 square hectometer (sq. Hm.) 
= 1 square kilometer (sq. Km.) 


Land Measure 


325. The unit of land measure is the are. 

Table 

100 centares (ca.; = 1 are (a.) = 100 sq. m. 

100 ares = 1 hectare (Ha.) = 10000 sq. m. 


150 


DENOMINATE NUMBERS 


Cubic Measure 

326. The unit of volume is the cubic meter. 

Table 

1000 cubic millimeters (cu. mm.) = 1 cubic centimeter (cu. cm.) 

1000 cubic centimeters = 1 cubic decimeter (cu. dm.) 

1000 cubic decimeters = 1 cubic meter (cu. m.) 

Table of Wood Measure 

327. The unit of wood measure is the stere. 

10 decisteres (ds.) = 1 stere (s.) = 1 cu. m. 

10 steres = 1 decastere (Ds.) = 10 cu. m. 

Measure of Capacity 

328. The unit of capacity for either solids or liquids is the liter, 
which is equal in volume to 1 cu. dm. 

Table 

10 milliliters (ml.) = 1 centiliter (cl.) 10 liters = 1 decaliter (Dl.) 

10 centiliters = 1 deciliter (dl.) 10 decaliters = 1 hectoliter (HI.) 

10 deciliters = 1 liter (1.) 10 hectoliters = 1 kiloliter (Kl.) 

For measuring liquids in ordinary quantities the liter is used. 
For minute quantities the centiliter or the milliliter is used. 

For measuring grain, vegetables, etc., the hectoliter is used. 

Measure of Weight 

329. The unit of weight is the gram, which is the weight of 1 
cu. cm. of distilled water in a vacuum, at its greatest density 
39.2°F.). It weight 15.4324 gr. 

Table 

10 milligrams (mg.) = 1 centigram (c.g.) 10 hectograms = 1 kilogram (Kg.) 

10 centigrams = 1 decigram (d.g.) 10 kilograms = 1 myriagram 

10 decigrams = 1 gram (g.) (Mg.) 

10 grams = 1 decagram (Dg.) lOmyriagrams = 1 quintal (Q.) 

10 decagrams = 1 hectogram (Hg.) 10 quintals = 1 tonneau (T.) 

Notes. (1) The gram is used in weighing precious metals, drugs, letters, 
etc. 

(2) The kilogram is used for weighing ordinary merchandise, except in large 
quantities, for which the quintal or tonneau is used. 


TABLES OF EQUIVALENTS 


151 


330. The legal equivalent values of the units of both the 
English and the metric systems are given in the following: 


1 inch = 2.54 centimeters 
1 foot = .3048 of a meter 
1 yard = .9144 of a meter 
1 rod = 5.029 meters 
1 mile = 1.6093 kilometers 


sq. inch 
sq.foot 
sq. yard 
sq. rd. 
acre 
sq. mile 


TABLES OF EQUIVALENTS 

Linear Measure 

1 centimeter = .3937 of an inch 
1 decimeter = .328 of a foot 
1 meter = 1.0936 yards 
1 dekameter = 1.9884 rods 
1 kilometer = .62137 of a mile 

Surface Measure 


1 cu. inch 
1 cu. foot 
1 cu. yard 
1 cord 


1 dry quart 
1 liquid quart 


6.452 sq. centimeters 

1 sq. centimeter 

= .155 of a sq. inch 

.0929 of a sq. meter 

1 sq. decimeter 

= .1076 of a sq. foot 

.8361 of a sq. meter 

1 sq. meter 

= 1.196 sq. yards 

25.293 sq. meters 

1 are 

= 3.954 sq. rods 

40.47 ares 

1 hectare 

= 2.471 acres 

259 hectares 

1 sq. kilometer 

= .3861 of a sq. mile 

Cubic 

Measure 


16.387 cu. centimeters 

1 cu. centimeter 

= .061 of a cu. inch 

28.317 cu. decimeters 

1 cu. decimeter 

= .0353 of a cu. foot 

.7646 of a cu. meter 

1 cu. meter 

= 1.308 cu. yards 

3.624 steres 

1 stere 

= .2759 of a cord 

Measures 

of Capacity 


= 1.101 liters 

1 liter = .908 of a dry quart 


= .9463 of a liter 


1 liquid gallon = .3785 of a decaliter 
1 peck = .881 of a decaliter 

1 bushel = .3524 of a hectoliter 


1 liter 
1 decaliter 
1 decaliter 


= 1.0567 liquid qt. 
= 2.6417 liquid gal. 
= 1.135 pecks 


1 hectoliter = 2.8377 bushels 


Measures of Weight 

1 grain, Troy = .0648 of a gram 
1 ounce, Troy =31.104 grams 
1 ounce, avoir. = 28.35 grams 
1 lb., Troy = .3732 of a kilogram 
1 lb. avoir. = .4536 of a kilogram 


1 gram = 15.432 grains, Troy 
1 gram = .03215 of an oz. Troy 

1 gram = .03527 of an oz. avoir. 

1 kilogram = 2.6791b. Troy 
1 kilogram = 2.2046 lb. avoir. 


1 ton (short) = .9072 of a tonneau or ton 1 tonneau = 1.1023 tons (short) 

Convenient Equivalent Values 

1 cu. cm. of water = 1 ml. of water, and weighs 1 gram = 15.432 gr. 

1 cu. dm. of water = 11. of water, and weighs 1 Kg. = 2.2046 lb. 

1 cu. m. of water = 1 Kl. of water, and weighs 1 tonneau = 2204.6 lb. 


152 


DENOMINATE NUMBERS 


REDUCTION OF DENOMINATE NUMBERS 

331. Reduction of denominate numbers consists in changing 
the form or denomination of a number without changing its value. 

332. Reduction descending consists in changing denominate 
numbers from higher to lower denominations. 

333. Reduction ascending consists in changing denominate 
numbers from lower to higher denominations. 


REDUCTION DESCENDING 
334. Reduce 4 lb. 8 oz. 16 pwt. 14 gr. to grains. 


4 lb. =4X12X20X24 gr. = 23,040 gr. 
8 oz. = 8X20X24 gr.= 3,840 gr. 
16 pwt. = 16X24 *r. = 384 gr. 

14 gr. = 14 gr. 


4 lb. 8 oz. 16 pwt. 14 gr. =27,278 gr. 
the product of 4 X 12 X 20 X 24 gr. which gives 23,040 gr., etc. 


Reduce each denomina¬ 
tion to grains separately. 
Since each pound contains 
12 oz., each ounce contains 
20 pwt., and each penny¬ 
weight contains 24 gr., the 
number of grains in 4 lb. is 


PROBLEMS 


335. Reduce to lower denominations: 


1. 4 bu. 2 pk. 6 qt. 1 pt. 3. 15 cu. yd. 21 cu. ft. 1276 cu. in. 


2. 8 rd. 4 yd. 2 ft. 8 in. 4. 4 Dm. 8 m. 7 dm. 5 cm. 

5. A man buys 2 bu. 2 pk. of chestnuts at $3.00 per bushel 

and sells them at 10 £ a pint. How much does he gain? 

6. A cask of molasses containing 48 gal. cost 32 i a gallon and 

sold for 15 £ a quart. Find the gain. 


7. If a drug costs $1.50 per kilogram and sells for 5ff a deca¬ 
gram, what is the profit? 

8. Reduce f of a day to hours, minutes, etc. 

4X24 hr. = * 1 2 * * * * 7 8 9 10 - hr. = 14- hr. In f of a day there are f of 24 hr., or 


f X 60 min. = 24 min. 
14 hr. 24 min. 


14 1 hr. In f of an hour there are | of 60 
min., or 24 min. Hence, f of a day = 
14 hr. 24 min. 


9. Reduce f orf a bushel to lower denominations. 

10. A fence is to be f of a mile long. How many boards 12 ft. 
long will be required if the fence is 4 boards high? 



REDUCTION OF DENOMINATE NUMBERS 


153 


REDUCTION ASCENDING 


336. Reduce 573 in. to higher denominations. 


12)573 


3) 47, no. of ft. —9 in. 
5§) 15, no. of yd.—2 ft. 

yd. 


First divide by 12 because there are 
12 in. in a foot. The result is 47 ft., and a 
remainder of 9 in. Next divide by 3 
because there are 3 ft. in a yard. The 
result is 15 yd. and a remainder of 2 ft. 


2, no. of rd. 

Finally divide by 5^ because there are 5£ yd. in a rod. The final result is 
2rd. 4 yd. 2 ft. 9 in. 

PROBLEMS 


337. Reduce to higher denominations: 


1. 2489 gills. 

2 . 15,729 grains (Troy). 

3. 198,726 seconds. 

4. 36,729 square inches. 

5. 9,287 inches. 

ll. Reduce 2 sq. ft. 36 sq. in, 


6. 4372 minutes. 

7. 38,476 seconds (of longitude). 

8. 48,792 centigrams. 

9. 53,986 deciliters. 

10 . 58,371 millimeters. 

to the fraction of a square yard. 


144 )36.00 number of sq. in. Divide the 36 sq. in. by 144 (number 
9) 2.25 number of sq. ft. of square inches in a square foot), which 
c , gives .25 of a square foot. Add the 2 sq. 

.25 num er o sq. y . an( j divide by 9 (the num ber of square 

feet in a square yard). The result is .25 of a square yard. 

12 . Reduce 2 ft. 8 in. to the fraction of a yard. 

13 . Reduce 5 oz. 8 pwt. 12 gr. to the fraction of a pound. 

14. Reduce 5 hr. 48 min. to the fraction of a day. 

15. Reduce 4 cwt. 50 lb. to the fraction of a ton. 

16. Reduce 48 sq. rd. 21 sq. yd. 7 sq. ft. to the fraction of an acre. 

17. Reduce 4 hr. 30 min. to the fraction of a day of 8 hr. 

18. A laborer worked 6 hr. 45 min. If a working day is 9 hr., 

what part of the day did he work? 

19. A garden contains 64 sq. rd. 4 sq. ft. 72 sq. in. What part 
of an acre is it? 

20 . A city lot 25 ft. by 100 ft. contains what part of an acre 
of land? 







154 


DENOMINATE NUMBERS 


COMPARISON OF MEASURES 


Weight 

338. For comparative tables of weights, see pages 143 and 151. 

339. 1 . Which is heavier, a pound avoirdupois or a pound Troy? 
an ounce avoirdupois or an ounce Troy? Explain. 


2. How do the pound and ounce of Troy and apothecaries’ 
weights compare? 


340. How many pounds Troy are equal to 12 lb. avoirdupois? 


175 


n X 7000 _ 175 
3700 12 1412 

Ui 


lb. Troy. 


12 


In one pound avoirdupois there are 
7000 gr.; in 12 lb. there are 12 times 
7000 gr. Dividing by 5760 gr. in one 
pound Troy gives the number of Troy 
pounds, which is 14 T V 

The cancellation form of solution is 
best adapted to these problems. 


PROBLEMS 

341. 1. Change 16 lb. avoirdupois to Troy pounds. 

2. Change 25 lb. Troy to avoirdupois pounds. 

3 . Reduce 22 lb. avoidupois to apothecaries’ pounds. 

4 . How much is gained per pound by buying a drug at S3.60 
per avoirdupois pound, and selling it at 10 ^ per dram? 

5. A jeweler buys 3 lb. 8 oz. of gold and manufactures it into 
64 watch chains. What is the average weight of the chains? 

6. How many pounds are there in 26 Kg.? 

A kilogram weighs 2.2046 lb., and 26 

26X2.2046 lb. = 57.3196 lb. Kg. weigh 26 times 2.2046 lb., or 57.3196 

lb. 

7. Reduce 20 Kg. to pounds. 

8. Reduce 45 Kg. 5 Hg. to pounds. 

9 . Reduce 27 lb. to kilograms. 

10 . How many grams of pure gold will be required to make 24 
gold rings 18 karats fine each weighing 5 pwt. 8 gr.? 

11 . Find the gain on a carload of coal weighing 24 long tons 
bought at $6.50 per long ton, and sold at $10.25 per short ton. 



REDUCTION OF DENOMINATE NUMBERS 


155 


MONEY 


342. In dealing with other nations it is necessary to reduce 
foreign money to United States money, and vice versa. 


343. 1 . Change £228 16s. 8 d. to United States money. 

16s. ^ 20s. = £.8. 

228.8 X $4.8665= $1113.46. 

8 d. = 16 i. 


First reduce the 16s. to the decimal 
of a pound by dividing by 20, and add the 
result, £ .8, to the £228, making £228.8. 
Multiply the value of £1, $4.8665, by 
$Illo.40 -f- lo 0 $lllo.O<s. 228.8, and the result, $1113.46 is the value 
in United States money of £228 16s. Each penny of English money is worth 
two cents of our money; 8 d. is worth 16 Adding $1113.46 and $.16 gives 
$1113.62, the value of £228 16s. 8d. in United States money. 

2. Change $683.62 to English money. 

Divide by $683.62 by the value of 


$683.62 -4- $4.8665 = £140.4748. 
.4748 X 20s. = 9.496s. 

.496 X 12 d. = 5.952d. = 6 d. 
Therefore £140 9s. 6 d. 


£1, $4.8665. The result, £140.4748, 
is the value in pounds and decimal 
of a pound. To change the decimal 
part of a pound to shillings, mul¬ 
tiply by 20, the number of shillings 
in a pound. The result is 9.496s. To change the decimal part of a shilling to 
pence, multiply by 12, the number of pence in a shilling. The result is 
£140 9s. 6d. 


3. Change F. 7284 to United States money. 

79SA V IMCtt - 81 SinCe 1 franC eqUals $ - 193 ’ 7284 fnmC8 

X “ &14UO.S1. equal 7284 times $.193 or $1405.81. 

4. Change $575 to francs. 

There are as many francs in 
$575 "5“ $.193 = 2979.27, no francs. *575 as $ .193 is contained times 

in $575 or 2979.27 times. 

5. Reduce M. 760 to dollars. 

^ 1 mark is worth $.238. 760 marks are 

760 X $.238 - $180.88. worth 760 times $ >2 38, or $180.88. 

6. Change $875 to marks. 

There are as many marks 
_ . in $875 as $.238 is contained 

$875 ^ $.238 = 3676.47, no. of marks. timeg in $875> or 3676 .47 

times. 

In the following problems, use the par value of the foreign 
coins. For values of foreign coins, see page 399. 


156 


DENOMINATE NUMBERS 


PROBLEMS 

344. 1. Reduce £147 7s. hd to dollars 

2. Reduce £325 18s. 7 d. to dollars. 

3. Reduce £458 15s. 9 d. to dollars. 

5. Reduce $685 to English money. 

5. Reduce $796 to English money. 

6. Reduce $1200 to English money. 

7. Reduce $1825.50 to English money. 

8. Reduce £1650 12s. 6 d. to dollars. 

9. Reduce F. 598 to dollars. 

10. Reduce F. 1285 to dollars. 

11 . Reduce F. 3484 to dollars. 

12 . Reduce $1593 to francs. 

13. Reduce $3327.75 to francs. 

14. Reduce $5897.50 to francs. 

15. Reduce F. 7500 to dollars. 

16. Reduce F. 6488.5 to dollars. 

17. Reduce M. 1372 to dollars. 

18. Reduce M. 2468 to dollars. 

19. Reduce M. 3896 to dollars. 

20. Reduce $4385 to marks. 

21. Reduce $3681.50 to marks. 

22. Reduce $1782.75 to marks. 

23. Silk is quoted by a French firm at F. 8.25 per meter. A 
similar grade of silk can be bought from a German firm for M. 6J 
per meter. Reckoning francs and marks at par, which is the 
better offer, and how much better, on an importation of 15,000 
meters of silk? 

24. A certain make of Swiss watch is quoted at F. 125. An 
English firm offers a watch of equal value for £4 18s. Which 
offer is better and how much better on 8 doz. watches? 

25. If lead pencils of a certain grade costing 18s. a gross in 
England can be bought in France at F. 22 per gross, find the 
difference in the cost of 100 dozen. 


REDUCTION OF DENOMINATE NUMBERS 


157 


MISCELLANEOUS MEASURES 


345. For comparison of measures see pages 139 to 141. 

346. Find the weight of a barrel of water. 


3.5 77 

3X.0 X ni X 62.5 lb. 

mi 

m 


16843.75 lb. 
64 


= 263.18 lb. 


64 


A barrel is 31? gal. A gallon contains 231 cu. in. 31£ times 231 gives the 
number of cubic inches in a barrel. Dividing by 1728 cu. in. to the cubic foot 
gives the number of cubic feet in a barrel. Multiplying by 62£ lb. to the cubic 
foot gives the weight of a barrel of water as 263.18 lb. 


PROBLEMS 


347. 1 . Find the weight of 25 gal. of water. 

2. How much does a 10-qt. pail of water weigh? 

3. A can of water weighs 45 lb. 10 oz. If the can weighs 8 lb., 
find the quantity of water in gallons. 

4. How many liquid gallons are equal in volume to 24 dry 
gallons? 

5. If a pail contains 10 qt. of berries, how many quarts of milk 
will it hold? 

6. A merchant bought cloth at 22 1 per meter and sold it at 30 <k 
per yard. Find his gain on 160 m. 


40 

m X 39.37 
30 


1574.8 

9 


= 174.97^, 


no. yd. 


9 

160 X $.22 = $35.20, cost. 

174.97^ X $.30 = $52.49, selling price. 

$52.49 - $35.20 = $17.29, gain. 

Change the 160 m. to yards by multiplying 160 by 39.37 in. in a meter, and 
dividing by 36 in. in a yard. The number of yards is 174.97f. 

The gain is found by taking the difference between the cost and the selling 
price. 

7. Change 48 m. to yards. 

8. Change 120 m. to yards. x 

9. Change 45 Km. to miles, rods, etc, 





158 


DENOMINATE NUMBERS 


10 . Change 12 mi. to kilometers. 

11. An express train in France runs at a speed of 76 Km. per 
hour. In the United States an express train runs 50 mi. per hour. 
Which train makes the better time, and how many miles per hour? 

12. If cloth costs 40 £ per meter and sells for 50 ^ a yard, find 
the gain on 348 m. 

13. Find the difference between 384 m. and 406 yd. 

14. How many square yards are equal to 450 sq. m.? 

15. Change 12 gal. to liters. 

12 x 4 qt. = 48 qt. 12 gal- = 48 qt. 1 1. = 1.0567 qt. In 

48 i 0567 = 45 42 no. liters 48 qt* there are as many liters as 1.0567 

is contained times in 48, or 45.42 1. 


16. How many decaliters are there in a barrel of vinegar? 

17. A farmer is offered $ .90 a bushel, or $2.50 a hectoliter, for 
his wheat. Does he gain or lose, and how much, on 480 bu. by 
accepting $ .90 a bushel? 


18. How many kilograms are there in a barrel of flour (196 lb.)? 


19. One wholesaler offers sugar at 5 £ a pound, and another 
at 11 i a kilogram. Which offer is the better, and how much, 
for the purchaser, on 20 bbl. averaging 350 lb. each? 


ADDITION OF DENOMINATE NUMBERS 

348. Add 6 gal. 3 qt. 1 pt. 3 gi\, 5 gal. 2 qt. 1 gi., 7 gal. 3 qt. 
1 pt. 2 gi., 3 gal. 1 qt. 1 pt. 2 gi. 

gi- 

3 Arrange the addends so that like units are in the 

1 same column. Add the lowest denomination first. 

2 The total number of gills is 8. But 8 gi. equals 2 
2 pints and no gills over. Write 0 in the answer place 

— under the gills’ column. Carry the 2 pt. to the 
23 3 1 0 pint column and add. The total is 5 pt., which 

equals 2 qt. and 1 pt. Write 1 in the pint column 
in the answer. Next add the quart column, carrying the 2 qt. from the pint 
column. The total is 11 qt., which is equal to 2 gal. and 3 qt. over. Write 3 
in the quart column. Carry the 2 gal. and add the last column. The total is 
28 gal., 3 qt., 1 pt. 


gal. qt. pt. 

6 3 1 

5 2 0 

7 3 1 

3 1 1 



ADDITION OF DENOMINATE NUMBERS 


159 


PROBLEMS 

349. l. Add: 5 rd. 4 yd. 2 ft. 6 in., 11 rd. 2 yd. 1 ft. 8 in., 7 rd. 
3 yd. 10 in., 12 rd. 1 yd. 2 ft. 

2. Add: 5 lb. 4 oz. 16 pwt. 12 gr., 2 lb. 9 oz. 10 pwt. 14 gr., 9 
lb. 8 oz. 18 pwt. 16 gr., 7 lb. 5 oz. 12 gr. 

3. Add: 5 A. 44 sq. rd. 15 sq. yd. 7 sq. ft. 98 sq. in., 13 A. 58 
sq. rd. 18 sq. yd. 7 sq. ft. 56 sq. in., 9 A. 68 sq. rd. 24 sq. yd. 5 
sq. ft., 17 A. 120 sq. rd. 22 sq. yd. 106 sq. in. 

4. A rectangular field is 26 rd. 2 yd. 1 ft. long, and 18 rd. 4 yd. 
2 ft. wide. What length of wire will be required for a fence five 
wires high? 

5. A farmer sold 4 loads of potatoes containing 48 bu. 24 lb., 51 
bu. 26 lb. 53 bu. 44 lb., and 43 bu. 50 lb. respectively. What 
was the total value at 75 i a bushel? 

6 . A milk dealer bought 6 gal. 2 qt. 1 pt. of milk from one man, 
8 gal. 3 qt. from another, and l5 gal. 2 qt. from a third. At 12 
a gallon, what did it cost him? 

7. Add: 8 Km. 7 Hm. 5 m. 6 dm., 5 Km. 4 Dm. 4 m., 3 Km. 
7 m. 9 dm. 

8. Find the sum of 7 Kg. 5 Hg. 4 Dg. 7 g., 9 Kg. 8 Dg. 4 g. 
5 dg., 7 Kg. 8 Hg. 3 g. 8 dg., 2 Hg. 5 Dg. 6 g. 8 dg. 


SUBTRACTION OF DENOMINATE NUMBERS 


350. From 16 rd. 4 yd. 2 ft. 4 in. take 12 rd. 5 yd. 2 ft. 8 in. 


rd. yd. ft. in. 

16 4 2 4 

12 5 2 8 

3 3® 2 8 

\ =1 6 

3 4 12 


Arrange the numbers with like denominations in 
the same column. Since 8 in. cannot be taken from 
4 in., “borrow” 1 ft., reduce it to inches, and add to 
the 4 in. (12 +4 = 16). Now take 8 in. from 16 in. 
leaving 8 in. Write 8 in the remainder. 1 ft. was 
“borrowed,” leaving 1 ft. 2 ft. cannot be taken from 
1 ft. “Borrow” 1 yd., reduce to feet and add to the 
1 ft. (3-4-1 =4). 2 from 4 leaves 2 for the re¬ 


mainder. Next borrow 1 rd., reduce and add to the yards (5§ + 3 = 8^). 
5 from 8§ leaves 3^ as a remainder. 12 from 15 leaves 3. To eliminate 
the £ yd. in the remainder, reduce it to feet and inches, yd. = 1 ft. 6 in.), 
and add as illustrated. 




160 


DENOMINATE NUMBERS 


351. Subtract: 

da. hr. min. sec. 

1. 5 10 26 32 

3 12 48 16 

cd. cu.ft. 

4. 28 48 

14 72 


PROBLEMS 


bu. pk. qt. pt. 

2. 16 3 4 0 

8 3 6 1 

Km. Hm. Dm. m. 

5. 8 4 3 5 

6 7 5 9 


£ s. d. far. 

3. 28 11 6 2 

14 14 8 3 

cu. m. cu. dm. cu. cm. 

6. 716 127 243 
285 458 759 


7. From a cask containing 56 gal. 2 qt. 1 pt. of vinegar, 27 gal. 
3 qt. were drawn out. How much vinegar remained? 

8 . A field contains 38 A. 116 sq. rd. 18 sq. yd. 9 A. 28 sq. rd. 
are planted to corn. 6 A. 96 sq. rd. 20 sq. yd. are planted to pota¬ 
toes, and the remainder of the field is meadow. Find the area of 
the meadow. 

9. From J bu. take 2| pk. 


MULTIPLICATION OF DENOMINATE NUMBERS 

352. Multiply 5 lb. 7 oz. 8 pwt. 15 gr. by 6. 

Arrange the numbers as shown in the illustration. 
Begin with the lowest denomination. The first 
product is 90 gr. Dividing by 24 gr. (in a pwt.) 
gives 3 pwt. 18 gr. Write 18 gr. and carry the 3 
pwt. The next product is 51 pwt. which is equal to 
2 oz. and 11 pwt. Write 11 and carry 2. Continue 
this process until each denomination in the multipli¬ 
cand has been multiplied. 

PROBLEMS 

353. 1 . Multiply 5 gal. 3 qt. 1 pt. 2 gi. by 7. 

2. Multiply 6 da. 5 hr. 14 min. 17 sec. by 9. 

3. Multiply 64 sq. rd. 13 sq. yd. 7 sq. ft. 54 sq. in. by 8. 

4. Multiply 4 mi. 124 rd. 3 fc yd. 2 ft. 9 in. by 15. 

5. Multiply 2 lb. 5 oz. 6 dr. 2 sc. 8 gr. by 9. 

6. A druggist bought 7 bottles of sulphuric acid, each containing 
8 lb. 6 oz. (Avoir.). Find the total weight. 


lb. oz. pwt. gr. 

5 7 8 15 
6 

33 44 51 90 
33 8 11 18 










DIVISION OF DENOMINATE NUMBERS 


161 


7. If 1 bag holds 2 bu. 1 pk. 4 qt. 1 pt. of grain, how much will 
36 such bags hold? 

8. A jeweler bought a dozen silver spoons, each weighing 2 oz. 
8 pwt. 16 gr. At 5 £ a pennyweight, what did they cost? 

9. If 1 cask of molasses contains 43 gal. 3 qt. 1 pt., find the cost 
of 12 casks at 30 per gallon. 

10 . The drive wheel of an engine is 5 yd. 2 ft. 4 in. in circum¬ 
ference, and makes 4 revolutions per second. What fraction of a 
mile does the engine travel per minute? What is the speed per 
hour? 

DIVISION OF DENOMINATE NUMBERS 


354. Divide 4 bu. 3 pk. 1 qt. by 3. 


bu. 

3)4 


pk. 

3 


Arrange the dividend and divisor as illustrated. 
The first partial quotient is 1, with a remainder of 
12 3 1 bu. 1 bu. equals 4 pk. Adding the 3 pk. gives 

7 pk. Dividing by 3 gives 2 as the next part of the 
quotient, with a remainder of 1 pk. 1 pk. equals 8 qt., which with the 1 
quart makes 9 qt. Dividing by 3 gives 3, the final figure of the quotient. 
The result is 1 bu. 2 pk. 3 qt. 


PROBLEMS 

355. l. Divide 4 da. 5 hr. 27 min. 18 sec. by 6. 

2. Divide 15 lb. 8 oz. 5 dr. 2 sc. by 4. 

3. Divide 2 bbl. 17 gal. 3 qt. 1 pt. 2 gi. by 7. 

4. Divide 17 cu. yd. 1264 cu. in. by 16. 

5. A field containing 12 A. 106 sq. rd. 14 sq. yd. was cut up 
into 172 city lots after a deduction of 2 A. 127 sq. rd. 8f sq. ft. 
was made for the streets. What was the area of each lot in square 
feet? 

6. If 18 horses in 60 da. eat 624 bu. 1 pk. 4 qt. of oats, how 
much does each horse eat each day? 


VAN TUYDiS NEW COMP. AR.—11 



INVOLUTION AND EVOLUTION 


INVOLUTION 

356. A power of a number is the product obtained by multiply¬ 
ing the number by itself. 16 is a power of 4, because 4X4 = 16. 

357. Powers are named from the number of times a given factor 
is used to produce the power. 

Thus, 36 is the second power (called “square”) of 6, because 6 is used twice 
as a factor to produce 36. 

27 is the third power (called “cube”) of 3, because 3 is used three times as 
a factor to produce 27. 

256 is the fourth power of 4, because 4 is used four times as a factor to 
produce 256, etc. 

358. The number of times a given factor is to be used is shown 
by a small figure, called an exponent, written to the right of the factor. 

Thus in 5 2 , 2 is the exponent and shows that 5 is to be used twice as a factor; 

5 2 = 5 X 5 = 25. 

In the same manner 4 3 = 4 X 4 X 4 = 64. 

5 4 = 5X5X5X5 = 625, etc. 

359. Involution is the process of finding the power of a number. 

PROBLEMS 

360. 1-20. Find the square, cube, and the fourth power of all 
the numbers from 1 to 20 inclusive. 

21-40. Find the square, cube, and the fourth power of all the 
two-place decimals .01 to .20 inclusive. 


Find the square, cube, and the fourth power of: 


41. 

1 

2 

45. f 

49. | 

53. -re 

57. 

8 

9 

42. 

i 

46. i 

50. f 

54. 

58. 

T2 

43. 

2 

3 

47. | 

51. i 

55. f 

59. 

TIT 

44. 

1 

4 

48. f 

52. f 

56. f 

60. 

1 0 
1 7 


162 


EVOLUTION 


163 


EVOLUTION 

361. 1 . What number multiplied by itself will produce 9? 16? 
25? 36? 49? 64? 81? 

2 . What number used three times as a factor will produce 8? 
27? 64? 125? 

362. The root of a number is one of the equal factors of that 
number. Thus, 4 is a root of 16; 5 is a root of 125; etc. 

363. The square root of a number is one of the two equal factors 
which produce the number. 5 is the square root of 25. 

364. The cube root of a number is one of the three equal factors 
which produce the number. 6 is the cube root of 216. 

365. Evolution is the process of finding the root of a number. 

366. The radical sign (V) denotes that # root of a number is 
to be found. 

625 indicates that the square root is to be found. 

\/343 indicates that the cube root is to be found. 

367. The following diagram will illustrate the principle involved 
in finding the square root of a number. 

The square is 15 ft. on a side, and is divided 
into four parts as follows: 

1 square 10 ft. on a side 

= (10 ft.) 2 = 100 sq. ft. 

2 rectangles 10 ft. long and 5 ft. wide 

= 2 X (10 X 5) sq. ft. = 100 sq. ft. 

1 square 5 ft. on a side 

= (5 ft.) 2 = 25 sq. ft. 

Adding the several results gives 

[10 2 + 2 X (10 X 5) + 5 2 ] sq. ft. = 225 sq. ft. 

This result may be expressed as follows: 

The square of a number of two figures is equal to the square of the 
tens plus twice the product of the tens by the units plus the square of 
the units. 

By careful inspection and application of this principle, the 
square root of any number may be found. 


10 ft. 

50 sq. ft. 

25 « 
sq. ft. ^ 

4h 

50 £ 

100 sq. ft. © 

sq. ft. © 

10 ft. 

5 ft. 





164 


INVOLUTION AND EVOLUTION 


368. 1 . Find the square root of 225. 

V / 2 / 25 = 10 -r 5 = 15 

2 x 10 = 20 , trial divisor. 

1,00 

20 + 5 = 25, complete divisor. 

1,25 


1,25 


Beginning at units, divide the number into periods of two figures each. The 
largest perfect square in the left-hand period is 1. The square root of 1 is 1. 
Write, as part of the square root, 1 with a cipher to represent the remaining 

period in the number 225. The square of 
10 is 100, which subtract from 225, leaving 
125 as the new dividend. 

The accompanying diagram illustrates 
the condition of the problem at this point. 

The remaining 125 sq. ft. is represented 
by the two rectangles a and b, and by the 
square c. Each rectangle is 10 ft. long. 
Hence, both rectangles are 2 X 10 ft., or 
20 ft. long. 20 ft. is the trial divisor as 
shown in the solution. 125 -f- 20 = 6, but 
6 is too large because, when it is added to 
the 20 to complete the divisor, it makes the 
divisor too large. Try 5 as the quotient figure. Complete the divisor by 
adding the 5 to the 20, making 25. The 5 is the width of the rectangles a and 
b, and is the length of one side of the square c. The complete divisor repre¬ 
sents the total length of both rectangles a and b, and of the square c. 

Now multiply the total length, 25, by the width, 5, and the result, 125, is 
just equal to the remaining area of the large square. Hence the square root of 
225 is 15. 

Note. The student should observe that 

2 X (10 X 5) + 5 2 = [(2 X 10) + 5] X 5. 

2. Find the square root of 1489.96. 

Vl4'89'.96 = 38.6 


9 

2X 3 = 6 ° 589 

60 + 8 = 68 5 44 

2 X 38 = 76° 45 96 

760 + 6 = 766 45 96 


The preceding solution and explanation were made in full that the principle 
involved might be thoroughly understood. The actual work of finding the 
square root is reduced to the minimum in this solution. 


a 

c 

100 sq. ft. 

b 

10 ft. 












EVOLUTION 


165 


In dividing a number involving a decimal, into periods of two figures each, 
begin at the decimal point and point off both to the right and to the left. 

The largest perfect square in the left-hand period is 9, and its square root is 
3. Write 3 in the root and its square, 9, under the 14, and subtract. Bring 
down the next period. The new dividend is 589. Take twice the root already 
found, and annex one cipher, giving 60 as a trial divisor. Divide and obtain 
8 (9 is too large). Complete the divisor by adding 8 to the 60. Multiplying 
68 by 8 gives 544 to be subtracted from 589, leaving 45. Bring the next period 
down. Since this period is decimal, place a point in the root after the 8 to 
separate the integral and decimal parts of the root. 

Now proceed exactly as before. Take two times the root already found, and 
annex one cipher. The trial divisor is 760. On dividing, the next figure in the 
root is found to be 6. Add the 6 to the trial divisor, and multiply the sum by 
the same 6. There is no remainder; hence, the square root of 1489.96 is 38.6. 

Note. The number of figures in the root are equal to the number of 
periods in the power. 

The following solutions illustrate additional points: 

3 . Find the square root of 5.247 to three places of decimals. 

V5.24 / 70 / 00 / 00 = 2.2906 = 2.291 
4 


2X2=4° 124 

40 + 2 = 42 "84 

2 X 22 = 44° 40 70 

440 + 9 = 449 40 41 

2 X 229 = 458°° 29 00 00 

45800 + 6 = 45806 27 48 36 


Annex a sufficient number of ciphers to make four full periods to the right of 
the decimal point. Proceed with the solution as already explained. Observe 
that the third decimal figure in the root is 0. In such cases bring down 
another period, and annex another 0 to the trial divisor as illustrated. Find 
the root to the fourth place of decimals and then take the nearest third place 
as the root. 

4 . Find the square root of .000729._ 

V.00'07'29 = -027 

4 

2X2=4° 3 29 

40 + 7 = 47 3 29 

As the left-hand period is composed of ciphers, write a cipher in the root to 
the right of the decimal point. The next period is 07. Treat it as though the 
7 were an integer, and proceed as before. 







166 


INVOLUTION AND EVOLUTION 


5. Find the square root of f f f . 


V169 = 13 
V256 = 16 



1 3 


TTf* 


To find the square root of a common 
fraction, find the square root of the numer¬ 
ator for the numerator of the root, and the 
square root of the denominator for the 
denominator of the root. 


When the numerator and 
denominator of the fraction 
.612 are not perfect squares, re¬ 
duce the fraction to its deci¬ 
mal form and find the square 
root of the resulting decimal. 
The degree of accuracy of the 
result is determined by the 
number of places the result is 
carried out. 


369. Find the square root of the following: 


1 . 

289 

7. 

15.452* 

13. 

.144 

19. 8 . 

2. 

961 

8. 

18.215 

14. 

.576 

on 5 7 8 

20 • 

3. 

1764 

9. 

18.0625 

15. 

.8925 

2i. m 

5. 

2809 

10. 

54.76 

16. 

.78095 

22. 

4. 

4489 

11. 

156.25 

17. 

.0000138 

23. f 

6. 

11449 

12. 

139.052 

18. 

2 . 

24. \ 


The problems requiring the application of square root in their 
solution will be found in Mensuration, pages 173 to 182. 

* Find the root to the nearest third place of decimals in each imperfect 
square. 


6 . Find the square root of 


2X6= 12° 

120 + 1 = 121 

2 X 61 = 122° 
1220 + 2 = 1222 


V.37'50'00 = 
36 
150 
121 


29 00 
24 44 


PROBLE 






MENSURATION 

LINES AND ANGLES 


370. A line has only one dimension—length. straight Line 

371. Parallel lines are lines that are the same - 

distance apart throughout their length. Parallel Lines 


372. A horizontal line is a line parallel to the horizon. 

373. An angle is the opening between two 
lines that meet. 

374. A right angle is one of the equal angles 
formed when two stright lines cross, making 
four equal angles. 

375. A perpendicular line is one that forms 

one or more right angles with another line. -- 



376. A vertical line is one that makes a right 
angle with a horizontal line. 


Perpendicular Lines 


SURFACES OR AREAS 

377. Surfaces have two dimensions—length and breadth. 


378. A quadrilateral is a plane surface hav¬ 
ing four sides and four angles. 



Quadrilateral 


379. A parallelogram is a quadrilateral hav- / f 

ing its opposite sides parallel. 


167 












168 


MENSURATION 


380. A rectangle is a parallelogram having 
right angles. 


381. A square is a rectangle having four equal 
sides. 


Square 



Rectangle 


382. A diagonal is a straight line between opposite angles of a 
quadrilateral. 


383. A triangle is a plane surface having 
three sides and three angles. 

384. A right-angled triangle is one that has 
one right angle. 

Note. No triangle can have more than one right 
angle. The three angles of any triangle are equal to 
two right angles. 



385. The hypotenuse of a right-angled triangle is the side 
opposite the right angle. 


386. An isosceles triangle is one that has two 
sides and two angles equal. 



387. An equilateral triangle is one that has 
all sides and angles equal. 


Isosceles Triangle 



Equilateral Triangle 












SURFACES OR AREAS 


169 


388. The base of any plane figure is the side on which it is 
assumed to stand. 

389. The altitude of any figure is the perpendicular distance 
from the base to the highest point opposite. 

390. A polygon is a plane surface bounded by straight lines. 
Polygons derive their names from the number of their sides; as, 
pentagon, hexagon, heptagon, octagon, etc. 

ooooo 

Hexagon "Heptagon Octagon Nonagon Decagon 



Pentagon 



391. A circle is a plane surface bounded by a curved line, every 
point of which is equally distant from the center of the circle. 

392. The circumference of a circle is its 
boundary line. 

393. The diameter of a circle is a straight 
line through its center terminating at each end 
in the circumference. 

394. The radius of a circle is one-half the 
diameter. 

395. The perimeter of a plane figure is the distance around it. 

396. The area of a plane figure is the number of square units 
within its boundary line. 

397. To find the area of any parallelogram. 

1 . Find the area of a rectangle 7 rd. long 
and 5 rd. wide. 

By drawing lines as shown in the diagram the sur¬ 
face of the rectangle is divided into square rods. 

Since the rectangle is 7 rd. long, there are 7 sq. rd. in the lower row of 
squares. There are 5 rows of squares, because the rectangle is 5 rd. wide. If 
in one row there are 7 sq. rd., in 5 rows there are 5 times 7 sq. rd., or 35 sq. rd., 
the required area. 

2 . Find the area of a parallelogram the base of which is 9 rd. 
and the altitude, 6 rd. 


7 rods 
















170 


MENSURATION 


Represent the parallelogram, as in the diagram, by abed. It is easily proved 
by Geometry that if that part of the parallelogram indicated by the triangle 
dec were cut off and placed in the position of the triangle 
afb , the resulting figure would be the rectangle afed, 
having a base/e equal to the base be of the parallelogram. 

The area of the rectangle afed is 6 times 9 sq. rd., or 
54 sq. rd., as explained in the preceding example. But 
the areas of the rectangle and of the parallelogram are equal. Hence the area 
of the parallelogram is equal to 6 times 9 sq. rd., or 54 sq. rd. 

The area of any parallelogram is equal to the product of its base by 
its altitude. 

Note. In this and similar rules the product of the dimensions means the 
product of the numbers that represent them, when they are expressed in like 
units. 

PROBLEMS 

398. Find the perimeter and the area of the following rectangles: 

(Make a diagram of each.) 


1 . 

40 

ft. by 36 ft. 

6. 

7 rd. by 14 yd. 

2. 

15 

yd. by 14 yd. 

7. 

9 yd. by 8ft. 6 in. 

3. 

18 

yd. square. 

8. 

128.5 rd. by 83.75 rd. 

4. 

20 

rd. by 18 rd. 

9. 

\ mi. by 76 rd. 

5. 

56 

rd. by 48 rd. 

10. 

8 Hm. by 5 Hm. 6 Dm. 


11. How many acres are there in a rectangular field \ mi. long 
and 66 rd. wide? 

12 . A field in the form of a rectangle contains 1 A. What length 
of fence is required to inclose it if the perpendicular distance 
between its sides is 10 rd.? 

Note. Since the product of the base by the altitude is equal to the area, 
the base may be found by dividing the area by the altitude. What principle 
applies? Before dividing, reduce the area (1 A.) to square rods. 

13. Find the perimeter of a square field containing 10 A. 

How do the base and altitude of a square compare in length? 

By what process may one of two equal factors of a product be found? 

14. A rectangular field contains 36 A. Find the cost at 75 £ a 
rod of building a fence around it if the width is f of the length. 




SURFACES OF AREAS 


171 


15 . Find the cost of a cement sidewalk 80 ft. long and 4 ft. 
6 in. wide, at SI.25 per square yard. 

16 . How many paving blocks 1 ft. long and 5 in. wide will be 
required for a street 1 mi. long and 40 ft. wide? 

17 . At 27§ ji per square yard, find the cost of painting the four 
walls of a room 12 ft. long, 9 ft. 6 in. wide, and 8 ft. high. 

18 . The Hudson Terminal Building in New York, N. Y., con¬ 
tains 4000 offices, and has a floor space of 27 A. Find the average 
floor space per office. 

19 . A walk 6 ft. wide is built along two sides of a corner lot 80 
ft. square. Find the area of the walk: (a) if it is laid on the lot; 
(5) if it is laid on the outside of the lot. 

399. To find the area of any triangle. 

1. Find the area of a triangle having a base of 12 ft. and an 
altitude of 8 ft. 

12 x g How does the trian- 

-=48,no.sq.ft.,area, gle abd compare in size 

2 with the triangle bed ? 

What name is given to the figure abed ? Note that the 
diagonal bd divides the parallelogram abed into two 
equal parts. Since the area of the parallelogram is 
equal to the product of its base by its altitude, the area of the triangle bed is 
equal to one half the product of the base by the altitude. 

The area of any triangle is equal to one half the product of its base 
by its altitude. 

To find the area of a triangle when the three sides are given. 

It sometimes occurs that the altitude of a triangle is not known, 
but the length of each side is given. In such cases find the area 
as in the following example. 

2. Find the area of a triangle the sides of which are 18 ft., 
25 ft., and 31 ft. 

18 + 25 + 31 = 74 37 X 19 X 12 X 6 - 50616. 

| of 74 = 37 a/ 50616 = 224.98, no. sq. ft. 

37 - 18 = 19 

37 - 25 = 12 

37 - 31 = 6 







172 


MENSURATION 


Find one half the sum of the three sides, which is 37 ft. From 37 ft. sub¬ 
tract each side of the triangle, leaving 19,12, and 6 ft., respectively. Multiply 
the half sum, 37, and all the remainders, 19, 12, and 6, together, obtaining a 
product of 50,616. Extract the square root of 50,616, and the result, 224.98, 
is the number of square feet in the area. 

Note. Any triangle having sides in the ratio of 3, 4, and 5 is a right-angled 
triangle. 

3 . Make a diagram of a right- 
angled triangle having a base of 24 
rd., and an altitude of 16 rd., to the 
scale of iV' to a rod. 

Since the diagram is to be on the scale of 
tV' to a rod, the base, 24 rd., will be repre¬ 
sented by a line of an inch, or 1| in. long, 
and the altitude by a line 1 in. long. The 
triangle is completed by drawing the hypotenuse. 

PROBLEMS 

400. Make a diagram, and find the area, of triangles having 
dimensions as follows: 

1. Base 56 ft., altitude 32 ft. (Scale J" to a foot.) 

2. Base 48 yd., altitude 29 yd. (Scale to a yard.) 

3 . Base 64 rd., altitude 38 rd. (Scale §" to a rod.) 

5 . Base 88 rd., altitude 75 rd. (Scale iV' to a rod.) 

4 . Base 19 in., altitude 13 in. 

6. With sides 18 ft., 21 ft., and 25 ft. (Scale J" to a foot.) 

7 . With sides 27 ft., 35 ft., and 40 ft. (Scale §" to a foot.) 

8. With sides 72 rd., 85 rd., and 91 rd. (Scale 1 V' to a rod.) 

9 . With sides 65 rd., 73 rd., and 85 rd. (Scale to a rod.) 

10. With sides 5 in., 6 in., and 7 in. 

11 . Find the value of a triangular field having a base of 64 rd. 
and an altitude of 48 rd. at $115 per acre. 

12. The gables of a house are 32 ft. wide, and the ridge of the 
roof rises 20 ft. above the foot of the rafters. How many square 
feet of boards are required to cover both gables? 

13 . The sides of a triangular garden measure 56 ft., 49 ft., and 
45 ft. respectively. How many square yards are there in its area? 






SURFACES OF AREAS 


173 


14 . The sides of an equilateral triangular field are 100 rd. each. 
How many acres are there in the field? 

15 . A house 28. ft. wide has rafters on one side of the roof 19 ft. 
long, and on the other side 17 ft. long. If the foot of the rafters is 
18 f t. above the cellar wall, find the area of both gable ends of the house. 


Find the altitude of the following triangles: 

16 . Area 180 sq. ft., base 20 ft. 

Compare with note, page 170. 

17 . Area 270 sq. yd., base 45 yd. 

18 . Area 6 A., base 60 rd. 

19 . Area 15 A., base 100 rd. 


Find the base of the following triangles: 

20. Area 120 sq. in., altitude 15 in. 

21. Area 360 sq. yd. altitude 120 ft. 

22. Area 4 A., altitude 176 yd. 

23 . Area 20 A., altitude 80 rd. 

24 . The area of a triangle is 600 sq. ft. The differences be¬ 


tween one half the sum of all 
the sides and each side are 10 
ft., 20 ft., and 30 ft., respect¬ 
ively Find the three sides of 
the triangle. 

401. To find any side of a 
right-angled triangle having 
two sides given. 

a. To find the hypotenuse. 

1. Find the hypotenuse of a 
right-angled triangle whose 
base and altitude are 8 ft. 
and 6 ft., respectively. 

82 = 64 
6 2 = 36 

82 + 6 2 = 64 + 36 = 100 
VIOO = 10, no. ft., hypotenuse. 



In the diagram it is seen that the 
square having the base of the triangle 
as the length of its sides contains 
8X8, or 64, sq. ft. Likewise, the 
































174 


MENSURATION 


square having the altitude of the triangle as the length of its sides contains 
36 sq. ft. By adding these two areas, 64 sq. ft. and 36 sq. ft., the result, 100 
sq. ft., is found to be equal to the area of the square having the hypotenuse of 
the triangle as the length of its sides. To find one side of a square, its area 
being known, extract the square root of its area. The square root of 100 is 10. 
That is, the length of the hypotenuse is 10 ft. 

To find the hypotenuse of a right-angled triangle , extract the square 
root of the sum of the squares of the base and the perpendicular. 


b. To find the base or the perpendicular. 

1. Find the perpendicular of a right-angled triangle whose 
hypotenuse is 15 yd. and base 12 yd. 


Since the sum of the squares 
15 = 225 of the base and the perpendicular 

12 2 = 144 equals the square of the hypote- 

15 2 — 12 2 = 225 — 144 = 81 nuse ( see illustration in the pre- 

*/o7 n ri j* i ceding example), it follows that 

V81 =9, no. ft., perpendicular. the difference between the square 

of the hypotenuse and the square of one of the sides is equal to the square of 
the other side. 15 2 — 12 2 = 81. Since 81 is the square of the perpendicular, 
the perpendicular is equal to the square root of 81, or 9. Therefore the per¬ 
pendicular is 9 ft. 


To find either side of a right-angled triangle , having the hypot¬ 
enuse and one side given, extract the square root of the difference of 
the squares of the hypotenuse and of the given side 


PROBLEMS 

402. Make a diagram of, and solve, triangles measuring as 
follows: 

1. Base 24 ft., perpendicular 18 ft. Find hypotenuse and area. 

2. Base 30 ft., perpendicular 22\ ft. Find hypotenuse and area. 

3. Perpendicular 18 ft., base 13 ft. 6 in. Find hypotenuse and 
area. 

4. Hypotenuse 45 yd., perpendicular 27 yd. Find base and 
area. 

5. Hypotenuse 13 ft., base 12 ft. Find perpendicular and area. 

6. Hypotenuse 25 rd., perendicular 7 rd. Find base and area. 

7. How long a ladder placed 9 ft. from the side of a building 
will reach a window 40. ft high? 


SURFACES OR AREAS 


175 


8 . A guy rope 60 ft. long is attached to the top of a pole 48 ft. 
high. How far from the foot of the pole can the rope be fastened? 
(No allowance for sag.) 

9 . The base of a square pyramid is 14 ft. on a side. The 
vertical height of the pyramid is 24 ft. Find the area of its four 
sides. 

10. The rafters on a house are 25 ft. long. If the ridge of 
the roof is 16 ft. above the foot of the rafters, how wide is the 
house? 

11. A room is 14 ft. long, 11 ft. wide, and 8 ft. high. Find the 
length of a diagonal from one of the upper corners to the opposite 
lower corner. 

12. A ladder 61 ft. long stands close against a wall. If the bot¬ 
tom of the ladder is drawn out 11 ft., how far will the top of the 
ladder be lowered? 

13 . A square park containing 10 acres has two gravel walks 
across it perpendicular to each other and to the sides of the park. 
If the walk is 8 ft. wide, find its area in square yards. 

CIRCLES 

403. To find the diameter or circumference 
of a circle. 

1. Find the circumference of a circle whose 
diameter is 18 ft. 

3.1416 X 18 ft. = 56.5488 ft., circumference. 

It has been found by accurate measurement that the 
circumference of a circle is 3.1416 times the length of the diameter. 

To find the circumference of a circle , multiply the diameter by 
3.1416. 

2. Find the diameter of a circle whose circumference is 78.54 ft. 

78.54 -4- 3.1416 = 25 ft., diameter. 

The circumference is 3.1416 times the diameter. 

The circumference divided by 3.1416 will give the diameter. (What 
principle applies?) 

Note. For approximate measurements, 3f may be used instead of 3.1416. 




176 


MENSURATION 


PROBLEMS 

404. Find the approximate and the accurate circumferences of 
circles having diameters as follows: 


1. 14 ft. 

5. 45 ft. 6 in. 

9. 6 ft. 5 in. 

2. 28 yd. 

6. 56 yd. 

io. 1 rd. 2 yd. 2 ft. 7 in. 

3. 35 in. 

7. 4 ft. 8 in. 

li. 112 rd. 

4. 42 ft. 

8. 5 ft. 3 in. 

12. 100 rd. 

Find the approximate and the accurate diameters of circles 

having the following circumferences: 


13. 11 ft. 

17. 110 yd. 

21. 51 yd. 1 ft. 

14. 66 ft. 

18. 140 yd. 

22. 53. ft. 2 in. 

15. 5 ft. 6 in. 

19. 55 ft. 

23. 1 mi. 

16. 33 rd. 

20. 7 ft. 4 in. 

24. 1 rd. 

25. A bicycle wheel is 2 ft. 4 in. in diameter. Over what distance 


will the rider have passed when the wheel has made 5000 rota¬ 


tions? 

26. A horse is tied to a post with a rope 30 ft. long. What is the 
circumference of the largest circle over which he can graze? 

405. To find the area of a circle. 

By examining the diagram in the margin it is seen that a circle is 
composed of a great number of small triangles whose bases form the 
circumference of the circle and whose vertices 
meet at the center of the circle. 

Note. While the bases of the triangles shown in the 
diagram are not straight, but curved, lines, yet the 
truth of the above statement is readily proved by 
Geometry. 

How may the area of a triangle be found? 

What is the shortest way to find the total area of 
three triangles having equal altitudes? 

What is the altitude of the triangles in the diagram? 

The sum of all their bases is equal to what line? 

How may the area of a circle be found? 

406. Find the area of a circle 18 ft. in diameter. 

First Solution 

3.1416 X 18 ft. = 56.5488 ft., circumference. 

18 ft. 4 = 4| ft., one half the radius. 

4§ X 56.5488 = 254.4696, no. sq. ft., area. 







SURFACES OF AREAS 


177 


Since the diameter of a circle is given, find the circumference by multiplying 
the diameter by 3.1416. The circumference is the sum of the bases of all the 
triangles in the circle. Find half the radius of the circle by dividing the 
diameter by 4. One half the radius is equal to one half the altitudes of the 
triangles. Multiply the circumference, 56.5488, by the half radius, 4j, and 
the result, 254.4696, is the number of square feet in the area. 


The labor of computing the area of a circle may be lessened 
by writing the solution in the cancellation form, and observing the 
resulting combinations of numbers. Thus, in the problem given 
above, we have: 

Second Solution 


.7854 

2.UZ0 X 18 x 18 
£ 


= .7854 x 182 = .7854 X D2. 


18 2 = 324. 

.7854 X 324 = 254.4696, no. sq. ft., area. 


Having the numbers written as illustrated, note that the numerator con¬ 
sists of 3.1416, which is the same for all circles, and of the diameter written 
twice. The denominator consists of 4, because half the radius is one fourth 
of the diameter, which is true for all circles. 

Now by canceling the 4 into 3.1416, the decimal .7854 is obtained. 

Therefore, by memorizing the decimal .7854, the area of a circle is readily 
found. 


To find the area of a circle multiply the square of the diameter (D 2 ) 
by .7854. 

In some problems the work may be still further abbreviated. 
The same problem will serve as an illustration. 


Third Solution 


9 9 

3.1416 x n x n 
£ 

2 


3.1416 X 9 2 = 3.1416 X R 2 . 


92 = 81; 3.1416 X 81 = 254.4696, no. sq. ft., area. 


Instead of canceling the 4 into 3.1416, divide the 18’s each by 2 (2 and 
2 being the factors of 4). The result is two 9’s. But 9 is the radius of the 
circle. Therefore, the area of a circle is equal to the square of the radius 
multiplied by 3.1416. 

VAN TUYL’S NEW COMP. AR—14 




178 


MENSURATION 


The area of a circle is equal to : 

(1) The circumference multiplied hy one half the radius, or, 

(2) The square of the diameter multiplied hy .7854 (} of 3.1416), or 

(3) The square of the radius multiplied by 3.1416. 

Practice alone can tell the student which rule is the best for 
the individual problem. In general, however, it may be said that, 
if both diameter and circumference are known, the first rule is 
good. 

If the diameter is a small number and can be squared mentally, 
the second rule is good. 

If the diameter is a number that is not readily squared mentally, 
and the radius can be so squared, the third rule is a good one. 


PROBLEMS 


407. 1 -26. Find the area of all the circles in the problems on 
page 176. 

For approximate areas, 3| may be used instead of 3.1416. 

408. To find the diameter or the circumference of a circle when 
the area is known. 

Find the diameter of a circle whose area is 201.0624 sq. rd. 

201.0624 -f- .7854 = 256, the square of the diameter. 

V256 = 16, no. rd., diameter. 

Since the square of the diameter multiplied by .7854 gives the area of the 
circle, the area divided by .7854 gives the square of the diameter, or 256. 
Extracting the square root of 256 gives 16, the number of rods in the diameter. 
(Prin. 14, page 117.) 

To find the diameter of a circle whose area is given, divide the area 
hy .7854 and extract the square root of the quotient . To find the 
circumference, multiply the diameter hy 3.1416. 


PROBLEMS 


409. Find the diameter and the circumference of circles whose 
areas are: 


1. 12.5664 sq. ft. 

2. 28.2744 sq. rd. 

3. 50.2656 sq. ft. 


4. 78.54 sq. yd. 

5. 113.0976 sq. yd. 

6. 153.9384 sq. rd. 


SOLIDS 


179 


7. 452.3904 sq. rd. 

8. 100 sq. rd. 


9. 400 sq. rd. 11. 15 acres. 

10. 1 acre. 12. 502.656 acres. 


13. 125.664 acres. 


14. 1000 acres. 


REVIEW OF PLANE FIGURES 
PROBLEMS 


410. l. How much more will it cost at SI.15 per rod to build a 
fence around 40 A. in the f orm of a square than in the f orm of a circle? 

2. Find the area of a circle that can be described with a radius 
of 85 ft. 

3. A rectangular field containing 32 A. is f as wide as it is long. 
What are its dimensions? 

4. An isosceles triangle has a base 10 ft. long, and an altitude 
of 12 ft. Draw a diagram and find the other sides of the triangle. 

5. An equilateral triangle is 12 ft. on a side. Find its altitude. 

6. A series of triangles having a common altitude of 9 ft. have 
bases of 4 ft., 6 ft., 3 ft. 6 in., 7 ft. 4 in., and 8 ft. 2 in., respec¬ 
tively. Find their total area. 

7. A hexagon is 6 ft. on a side. Find its area. 

8. If a walk around a court 200 ft. square occupies one fourth 
the area of the court, find the width of the walk. 

9. If a triangular area having sides 36 ft., 45 ft., and 60 ft. 
long, respectively, is paved at a cost of $2.60 per square yard, how 
much does it cost? 

10. A square park has a walk diagonally through it. If the 
length of the walk is 50 rd., find the area of the park. 


SOLIDS 


411. A solid is a figure having three dimen¬ 
sions—length, breadth, and thickness. 



412. A prism is a solid whose bases or ends are 
any similar, equal, and parallel plane figures, and 
whose lateral faces are parallelograms. 


Note. Prisms are named from the number of their 
sides—three sides, triangular , four sides, quadrangular or 
square , etc. 


Square Prism 























180 


MENSURATION 


413. A cube is a prism having six equal square 
faces. 



Cube 


414. A cylinder is a solid whose bases are equal, 
parallel circles, and whose lateral surface is a uni¬ 
form curve. 

Note. In this book the term “cylinder” is used for 
“circular cylinder.” 



415. The altitude of a solid is the perpendicular distance be¬ 
tween its bases. 


416. A pyramid is a solid having a polygon for 
a base and triangles for its sides. The vertices of 
the triangles form the vertex of the pyramid. 


Pyramid 

417. A cone is a solid having a circular base, 
and a lateral surface that tapers uniformly to a 
point. 

Note. In this book the term “cone” is used for “cir¬ 
cular cone.” 

Cone 

418. The slant height of a pyramid is the perpendicular dis¬ 
tance from one side of the base to the vertex. 




419. The slant height of a cone is the shortest distance from the 
circumference of the base to the vertex. 


420. To find the volume of a prism or a cylinder. 

1. Find the volume of a prism 8 ft. square and 10 ft. long. 
8 X 8 X 1 cu. ft. = 64 cu. ft. 

10X 64 cu. ft, s= 640 cu. ft., volume. 




















































SOLIDS 


181 


What is the area of a square 8 ft. on a side? 

What is the volume of a cube 1 ft. on each side? 

By consulting the diagram it is seen that one foot 
of the length of the prism contains 8 X 8, or 64, 
cubes, one foot on a side, that is, 64 cu. ft. If 1 
ft. of the length contains 64 cu. ft., the whole 
length will contain 10 times 64 cu. ft., or 640 
cu. ft. 

2. Find the volume of a cylinder 4 ft. 
in diameter and 8 ft. long. 

3.1416 X 2 X 2 X 1 cu. ft. = 12.5664 cu. ft. 

8 X 12.5664 cu ft = 100.5312 cu. ft., volume. 

As in the preceding example, one foot of the length of the cylinder contains 
as many cubic feet as the number of square feet in the area of the base. The 
area of the base is the area of a circle 4 ft. in diameter, which is equal to 3.1416 
times the square of the radius, or 12.5664. The entire cylinder contains 8 
times 12.5664 cu. ft., or 100.5312 cu. ft. 

To find the volume of a prism or of a cylinder, multiply the area of 
the base by the altitude and express the product in cubic units. 

PROBLEMS 

421. Find the volumes of the following solids: 

1. Square prism, base 6 ft. by 6 ft., altitude 7 ft. 

2. Rectangular prism, base 3 ft. by 4 ft., altitude 6 ft. 

3. Triangular prism, sides of base 4 ft., altitude 8 ft. 

4. Cylinder, diameter 5 ft., altitude 9 ft. 

5. Cylinder, diameter 5 ft. 6 in., altitude 10 ft. 

6. Cylinder, diameter 3 in., altitude 5 in. 

7. Hexagonal prism, side of base 3 ft., altitude 12 ft. 

8. Triangular prism, sides of base 3 ft., 4 ft., and 5 ft., altitude 9 ft. 

9. Triangular prism, sides of base 6 ft., 5 ft., and 5 ft., altitude 8 ft. 

10. Cylinder, diameter 16 ft., altitude 12 ft. 

11. A cylindrical water tank 25 ft. in diameter, and 30 ft. deep 
is full of water. How many tons of water are there? 

12. A block of marble is 6 ft. long, 3 ft. 6 in. wide, and 2 ft. 6 in. 
thick. What is its weight if marble is 2.7 times as heavy as water? 















182 


MENSURATION 


13. A rectangular tank is 12 ft. long, 4 ft. 9 in. wide, and 2 ft. 
3 in. deep. How long will it take to fill it at the rate of a gallon a 
minute? 

14. Find the dimensions of the base of a square prism whose 
altitude is 15 in., and whose volume is 540 cu. in. 

540 cu. in. -r- 15 = 36 cu. in., volume of base of the prism one 
inch high, a/36 = 6, no. in., one side of base. 

Since the area of the base multiplied by the altitude equals the volume, the 
volume divided by the altitude equals the area of the base, and since the base 
is square, its dimensions are found by extracting the square root of the area. 

15. The volume of a prism is 360 cu. ft. and its base is 4 ft. by 
5 ft. Find its altitude. 

16. A cylinder is 10 ft. long and its volume is 785.4 cu. ft. Find 
its diameter. 

17. The diameter of a cylinder is 20 inches, and its volume is 
1570.8 cu. in. Find its length. 

422. To find the lateral surface and the entire surface of a 
prism or of a cylinder. 

1. Find the lateral surface and the entire surface of a triangular 
prism, if its base is 4 ft. on a side, and its altitude is 8 ft. 

3 X4 ft. = 12 ft., perimeter of the base. 

8 X 12 sq. ft. = 96 sq. ft. area of lateral surface. 


6 X 2 X 2 X 2 = 48. 

V48 = 6.928, no. sq. ft., area of 1 
base. 

2 X 6.928 sq. ft. = 13.856 sq. ft., 
area of both bases. 

96 sq. ft. + 13.856 sq. ft. = 109.856 
sq. ft., area of entire surface. 

If a sheet of paper were cut so as just to 
cover all the faces of the prism, its form would 
be that shown by the dotted lines in the dia¬ 
gram. That part of the diagram which would 
cover the three lateral faces of the prism is a 
rectangle whose length is equal to the perim- 


(4 + 4 + 4) -*• 2 = 6. 
6-4 = 2. 





































SOLIDS 


183 


eter of the base of the prism, or 12 ft. The width of the rectangle is the altitude 
of the prism. The area of the rectangle is 8 X 12 sq. ft., or 96 sq. ft. 

To find the area of the entire surface the area of the two bases must be in¬ 
cluded. The bases are equilateral triangles 4 ft. on a side, whose areas are each 
6.928 sq. ft. (See page 171). The entire area, therefore, is the sum of 96 sq. 
ft. and 13.856 sq. ft. (both bases), or 109.856 sq. ft. 

2. Find the area of the entire surface of a cylinder 5 ft. in diam¬ 
eter and 10 ft. long. 

3.1416 X 5 ft. = 15.708 ft., circumference of base. 

10 X 15.708 sq. ft. = 157.08 sq. ft., lateral area. 

(5 2 ) sq. ft. X .7854 = 19.635 sq. ft., area of 1 base. 

2 X 19.635 sq. ft. = 39.27 sq. ft., area 
of both bases. 

157.08 sq. ft. + 39.27 sq. ft. = 196.35 
sq. ft., area of entire surface. 

As in the preceding example, a sheet of paper 
just large enough to cover the lateral surface of 
the cylinder would be in the form of a rectangle 
whose length is equal to the circumference of 
the cylinder and whose width is the altitude of 
the cylinder. The area of the lateral surface is 
found by multiplying the circumference (3.1416 
X 5 ft.) by the altitude of the cylinder. The 
area of the bases is the area of the two. circles whose diameters are 5 ft. (For 
explanation see pages 176, 177.) The lateral area, 157.08 sq. ft., plus the area of 
the two bases, 39.27 sq. ft., equals 196.35 sq. ft., the area of the entire surface. 

To find the lateral surface of a prism or of a cylinder, multiply the 
perimeter of the base by the altitude. 

To find the entire surface, add to the area of the lateral surface the 
area of the two bases. 

PROBLEMS 

423. l-io. Find the lateral, and the entire surface, of the cylin¬ 
ders and prisms mentioned in problems 1-10 on page 181. 

11. How many square yards of sheet iron would be required for 
a cylindrical reservoir 40 ft. in diameter and 40 ft. deep, if the top 
were open? 

12. Find the cost of painting the top, sides, and ends of a box 
8 ft. 4 in. long, 6 ft. 6 in. wide, and 4 ft. deep, at 20 i a square yard. 










184 


MENSURATION 


13. How many square yards of tin will be required to make 8 
dozen pails, without covers, 9 in. in diameter and 7\ in. deep, 
allowing 3 sq. ft. for seams and waste on each dozen pails? 

14. A wagon box is painted both inside and outside except the 
under side of the bottom. If it is 12 ft. long, 3 ft. 3 in. wide, and 
18 in, deep, find the area painted. (Make no allowance for 
corners.) 

15. A cylinder 6 ft. in diameter has 245.0448 sq. ft. in its entire 
surface. Find its altitude. 

(2X6X6) sq. ft. X .7854 = 56.5488 sq. ft., area of both ends. 

245.0448 sq. ft. — 56.5488 sq. ft. = 188.496 sq. ft., lateral area. 

6 ft. X 3,1416 = 18.8496 ft., circumference of cylinder. 

188.496 18.8496 = 10, no. ft., altitude. 

From the entire area deduct the area of both ends; the remainder is the 
the lateral area. Since the lateral area is the product of the circumference and 
altitude, the area divided by the circumference will give the altitude (Prin. 14, 
page 117). 

16. Find the altitude of a rectangular prism if its base is 5 ft. by 
8 ft. and its entire surface is 470 sq. ft. 

17. Find the diameter of a cylinder whose altitude is 11 ft. and 
whose lateral surface is 190.0668 sq. ft. 

18. A box 10 ft. square with a cover is made from 500 sq. ft. 
of 1 in. lumber. How deep is it? 
(Make no allowance for corners.) 

424. To find the volume of a pyra¬ 
mid or of a cone. 

Study the accompanying diagrams. 
Note the dimensions of the several 
forms. The contents of how many 
pyramids are being poured into one 
prism? What does that suggest as 
to the volume of a pyramid compared 
with a prism of the same dimensions? 

How many cones full are equal to 
one cylinder full? The volume of a 
cone is what part of the volume of a 
cylinder? 































SOLIDS 


185 


1. Find the volume of a pyramid 4 ft. square and 6 ft. high. 

4 X 4 sq. ft. = 16 sq. ft., area of the base. 

| of 6 = 2. 

2 X 16 cu. ft. = 32 cu. ft., volume of the pyramid. 

Since, as illustrated above, the volume of a pyramid is one third the volume 
of a prism of like dimensions, the volume of the pyramid is found by multiply¬ 
ing the area of the base by one third the 
altitude. 

2. Find the volume of a cone if its 
base is 6 ft. in diameter and its 
altitude is 8 ft. 

(3.1416 X 3 X 3) sq. ft. = 28.2744 
sq. ft., area of the base. 

8 X 28.2744 cu. ft. ono . . 

-= 75.3984 cu. ft., 

3 

volume of the cone. 

The volume of a cone is equal to one third 
the volume of a cylinder of like dimensions, 
and is found by multiplying the area of the 
base by one third the altitude. 

To find the volume of a pyramid or 
of a cone , multiply the area of the 
base by one third of the altitude and express the result as cubic units. 

PROBLEMS 

425. Find the volumes of the following pyramids and cones: 
Pyramids 

1. Square, base 6 ft. on a side, altitude 9 ft. 

2. Triangular, base 5 ft. on a side, altitude 10 ft. 

3. Rectangular, base 6 ft. by 8 ft., altitude 16 ft. 

Cones 

4. Diameter of base 5 ft., altitude 9 ft. 

5. Circumference of base 37.6992 dm., altitude 5 m. 

6. Circumference of base 24 ft., altitude 100 ft. 









186 


MENSURATION 


7. A square pyramid, the base being 8 ft. on a side, and the alti¬ 
tude 24 ft., has its top cut off 12 ft. from the apex, parallel to the 
base. The top part is what fractional part of the original pyramid? 

8. Find the diameter of the base of. a cone 15 ft. high whose 
volume is 251.328 cu. ft. 

251.328 v 5 = 50.2656, no. sq. ft., of base. 

50.2656 .7854 = 64. 

\/64 = 8, no. ft., diameter. 

Since the volume of a cone is equal to the area of the base multiplied by one 
third the altitude, the volume divided by one third the altitude must give the 
area of the base. Having the area of a circle, its diameter is found as explained 
on page 178. 

What principle applies? 

Will the same principle apply in the case of a pyramid? 

9. Find the dimensions of the base of a square pyramid if its 
volume is 128 cu. ft. and its altitude is 24 ft. 

10. A cone 42 in. high contains 1583.3664 cu. in. Find the 
diameter of the base. 

426. To find the lateral or convex surface of a pyramid or of a 

By inspecting the 
diagram in the mar¬ 
gin, it is seen that 
the lateral surface of 
a triangular pyramid 
consists of three tri¬ 
angles whose altitudes 
are the slant height 
of the pyramid and 
whose bases are equal 
to the sides of the 
base of the pyramid. 
How may the total 
area of several trian¬ 
gles having a common 
altitude be found? 
How may the lateral surface of a pyramid be found? Can the 
lateral surface of a cone be found in the same way? 


cone. 





SOLIDS 


187 


Find the lateral surface of a pyramid whose base is 4 ft. square 
and whose slant height is 8 ft. 

4 X 4 ft. = 16 ft., perimeter of the base. 

8 X 16 

——-sq. ft. = 64 sq. ft., area of lateral surface. 

The*.lateral surface of a square pyramid consists of four triangles having a 
common altitude, 8 ft. The sum of their bases is the perimeter of the base of 
the pyramid, or 16 ft. The area is found by multiplying the 16 ft. by 4 (half 
the slant height). 

To find the lateral surface of a pyramid or of a cone, multiply the 
perimeter of the base by one half the slant height. 

To find the entire surface add to the lateral surface the area of the 
base. 

PROBLEMS 

427. Find the lateral surface of pyramids and cones having 
dimensions as follows: 

Pyramids 

1. Square, side of base 5 ft., slant height 10 ft.* 

2. Square, side of base 4 ft., slant height 12 ft. 

3. Square, side of base 12 ft., slant height 27 ft. 

4. Square, side of base 3 m., slant height 1 Dm. 

Cones 

5. Diameter 5 ft., slant height 12 ft. 

6. Circumference 21.992 m., slant height 20 m. 

7. Circumference 31.416 ft., slant height 10 ft. 

8. Diameter 2 m. 4 dm., slant height 40 dm. 

9. A church spire is in the form of an octagonal pyramid, having 
a base 8 ft. on a side and a slant height of 75 ft. Find the Cv,st 
of roofing at 60 <f a square yard. 

10. The Pyramid of Cheops in Egypt is 746 ft. square, and 480 
ft. high. Find the area of its sides in acres, square rods, etc. and 
its volume in cubic yards. How many acres does its base cover? 

* Note the difference between “slant height” and “altitude.” 



188 


MENSURATION 


SIMILAR SURFACES 


428. How do the following pairs of plane figures differ? 


How does the diameter of the 
small circle compare with the 
diameter of the larter one? 




Compare the base of the larger 
square with the base of the smal¬ 
ler one. Compare their areas. 


Compare the bases and the 
areas of the two rectangles. 

l" ' 

429. Surfaces that have exactly the same shape, though they 
differ in size, are similar surfaces. 

430. Corresponding dimensions of similar surfaces are propor¬ 
tional. 

431. Principles: l. The areas of similar surfaces are to each 
other as the squares of their like dimensions. And, conversely , 

2. The like dimensions of similar surfaces are to each other as the 
square roots of their areas. 

432. l. A rectangle has a base of 6 ft. and an altitude of 5 ft. 
Find the base of a similar rectangle whose altitude is 15 

6 ft. : x ft. : : 5 ft. : 15 ft. 

6 X 15 

— - = 18, no. ft., base of similar rectangle. 

Since the like dimensions of similar rectangles are proportional, the base of 
the one rectangle bears the same ratio to the base of the other rectangle as the 
altitude of the one bears to the altitude of the other. By solving the resulting 
proportion, the base of the second rectangle is found to be 18 ft. 



3 













































SIMILAR SURFACES 


189 


2. A circle 20 rd. in diameter has an area of 314.16 sq. rd. Find 
the area of a circle 50 rd. in diameter. 


20 2 : 50 2 : : 314.16 sq. rd. : x sq. rd. 

78.54 

— - sc l- rd - = 25 X 78.54 sq. rd. = 1963J sq. rd. 

20X20 


The areas are proportional to the squares of the diameters. Hence the 
above proportion, the solution of which gives the area of the second circle as 
1963^ sq. rd. 


3. The areas of two similar triangles are 81 sq. in. and 169 sq. 
in. If the altitude of the smaller is 9 in., find the altitude of the 
larger. 


\/81 : \/169 : : 9 in. : x in. 

9 : 13 : : 9 in. : x in. 

9 in. X 13 

-~-= 13 in., altitude 

of the larger triangle. 


The like dimensions of the triangles are to 
each other as the square roots of the areas. 
Hence, the proportion as stated above. On 
solving, the altitude of the larger triangle is 
found to be 13 in. 


PROBLEMS 

433. l. A rectangular field is 60 rd. long and 49 rd. wide. Find 
the width of a similar field 75 rd. long. 

2. A field 80 rd. long contains 30 A. How many acres are there 
in a similar field 120 rd. long? 

3. A farm one half a mile long contains 66 A. How long is a 
similar farm of 264 A.? 

4. The area of a certain circle is 40 sq. ft. Find the area of 
another circle whose diameter is three times as great. 

6. A man desires to find the height of a certain tree. To find it 
he measures its shadow, and at the same time measures the shadow 
of a post 7 ft. high. If the shadow of the post is 10 ft. 6 in. long 
and the shadow of the tree 120 ft. long, what is the height of the 
tree? 

6. The areas of two circles are in the ratio of 16 to 25. If the 
diameter of the greater is 100 rd., what is the diameter of the lesser? 

7. If it costs $74.50 to build a fence around a square field of 
4 A., how much will it cost to build the same kind of a fence around 
a similar field containing 36 A.? 




190 


MENSURATION 


8. One sphere has twice the diameter of another sphere. What 
is the ratio of their surfaces? 

9. A given cube has 384 sq. ft. of surface. What is the surface 
of a cube whose edge is three times that of the given cube? 

SIMILAR SOLIDS 

434. How do the lengths, altitudes, and widths of these solids 

compare? 

They are similar solids 

because they have the same 
form though they differ in 
volume. 

Compare their volumes. 
The larger solid is how 
many times the smaller? 

435. Like dimensions of similar solids are proportional. 

436. Principles: l. The volumes of similar solids are to each 
other as the cubes of their like dimensions. 

2. The like dimensions of similar solids are to each other as the 
cube roots of their volumes. 

437. 1 . A solid is 8 ft. long, 6 ft. high, and 3 ft. wide. Find 
the height of a similar solid 10 ft. long. 








8 ft. : 10 ft. 
10 X 6 
8 


: 6 ft. :x ft. 


Since the like dimensions 
are proportional, the lengths 
of the two solids are in the 
ft. = 7^ ft.,height of similar solid. same ra ti 0 as the heights. 

Hence, the proportion is as 
stated. On solving, the height is found to be 7^ ft. 


2. The volume of a given solid 6 ft. long is 120 cu. ft. What is 
the volume of a similar solid twice as long? 


6 3 :12 s : : 120 cu. ft. : x cu. ft. 
12 X 12 X 12 X 120 


6X6X6 


cu. ft. = 960 cu. ft., volume of second solid 


Since one dimension of each solid and the volume of one of the solids are 
given, the cubes of the given dimensions are in proportion.to the volumes. 
On solving the resulting proportion, the volume of the second solid is found. 






























SPECIFIC GRAVITY 


191 


3. The diameter of a sphere is 3 in. What is the diameter of a 
sphere that is 27 times as large? 

vi : ^27 : : 3 : x. 

1 : 3 : : 3 : x. 

3X3 

—-— = 9. no. in., diameter of second sphere. 

Since the volumes are known, the ratio of their cube roots is equal to the 
ratio of the diameters. The cube root of 1 is 1, and of 27 is 3. Hence, the 
ratio of the diameters is as 1: 3. Therefore, the diameter of the second sphere 
is 9 in. 

PROBLEMS 

438. l. If a bin 10 ft. long holds 160 bu. of wheat how many 
bushels will a similar bin hold that is 20 ft. long? 

2. The volume of a sphere 1 in. in diameter is .5236 cu. in. Find 
the volume of a sphere 6 in. in diameter. 

3. The city water reservoir of a small village is 40 ft. in diame¬ 
ter, and 40 ft. deep. A new reservoir is necessary, and to provide 
for the future growth of the city, it is decided to erect a reservoir 
that will hold eight times as much water as the old one. If it is 
similar in form, what are its dimensions? 

4. A sphere having a 3-in. radius weighs 2 lb. 13 oz. What is 
the weight of a sphere of the same material whose radius is 12 in.? 

SPECIFIC GRAVITY 

439. Why does a piece of wood float in water? Why does a 
piece of iron sink? 

Name five substances lighter than water; five heavier than 
water. 

440. The ratio of the weight of a substance to the weight of an 
equal volume of water is the specific gravity of that substance. 
Thus, a cubic foot of cork weighs 15 lb., and a cubic foot of water 
weighs 62| lb. The ratio of 15 lb. to 62J lb. is expressed by the 

15 

fraction -= 15 -f- 62J = .24. Therefore, the specific gravity 

62 J 

of cork is .24. 



192 


MENSURATION 


441. In like manner find the specific gravity of each of the fol¬ 
lowing: 


Article Weight of Cu. Ft. Article Weight of Cu. Ft. 


l. Gold 

. . . 

1203| lb. 

8. Sugar 

100 

lb. 

2. Silver 

• . • 

6561 lb. 

9. Honey . 

91 

lb. 

3. Copper 

. 

549J lb. 

10. Milk. . . 

641 lb. 

5. Cast iron . 

450 lb. 

11. Butter . 

58j 

1 lb. 

5. Plate glass . 

1721 lb. 

12. Petroleum . 

55 

lb. 

6. Salt . 

. 

133| lb. 

13. Hickory, dry 

52i 

\ lb. 

7. Brick 

. . . 

125 lb. 

14. Alcohol . 

52i lb. 

Find the weight of a 

cubic foot of each of the following: 


Article 


Specific 

Article 

Specific 



Gravity 


Gravity 

15. Platinum . 

. 21.5 

20. Beeswax 

. 

.96 

16. Mercury 

. 14 

21. Lard . . 

. 

.947 

17. Brasj 

. . 

. 8 

22. Ice . . 

. 

.92 

18. Tin . 


. 7.19 

23. Turpentine . 

. 

.87 

19. Marble 

. . . 

. 2.72 

24. White pine, dry 

.40 

442. 

Table 

showing Specific Gravities 



Iridium . . . 

23.000 

Diamond . 

. . 3.550 Beeswax . 


. .960 

Platinum . . . 

21.500 

Plate glass . 

. . 2.760 Lard . . 


. .947 

Gold, pure . . 

19.258 

Marble . . 



. .942 

Mercury, pure . 

14.000 

Salt .... 

. . 2.130 Ice . . . 


. .920 

Silver .... 

10.500 

Brick . . . 



. .880 

Copper .... 

8.788 

Ivory . . . 



. .870 

Brass .... 

8.000 

Sugar . . . 


. .840 

Steel. 

7.840 

Honey . . 

. . 1.456 Alcohol . 


. .840 

Cast iron . . . 

7.200 

Milk . . . 


. .400 

Tin. 

7.190 

Water . . . 

. . 1.000 Cork . . 


. .240 


PROBLEMS 

443. l. A block of marble measures 4' by 2' by 1'6". How 
much does it weigh? 

2. Find the weight of a plate glass window 10' by 12' by J". 

3. A bar of cast iron is 21" long, 18" wide, and 1' thick. How 
much does it weigh? 






















SPECIFIC GRAVITY 


193 


4. A brick of pure gold is 8" X 4" X 2". Find its weight. 

5. A load of ice is 12 ft. long, 3 ft. 3 in. wide, and 2 ft. high. 
Allowing 8 cu. ft. for waste space, find its weight. 

6. Find the weight of 1000 brick 8" X 4" X 2" (to the nearest 
pound.) 

Any object floating in water displaces a weight of water equal to 
its own weight. 

Any object that sinks in water displaces a volume of water equal 
to its own volume. 

The weight of an object heavier than water is diminished by the 
weight of the water it displaces when weighed under water. 

7. What weight of water is displaced by a cubic foot of ice? 

8. A copper rod having a volume of l£ cu. ft. is under water. A 
pull of how many pounds is necessary to lift it? 

9 . A dry white pine plank 12 ft. long, 1 ft. wide, and 3 in. thick 
will support how many pounds without sinking in water? 

10. A vessel displaces 80,000 cu. ft. of water. What is its dis¬ 
placement? (Displacement means weight.) 

11. What is the specific gravity of a substance that floats with f 
of its volume under water? 

12. A block of ice 1 ft. thick, 8 ft. long, and 4 ft. wide will sup¬ 
port how many pounds in water? 

13. A can contains 10 gallons of milk. How much does the milk 
weigh? 

14. Into a jar full of water is placed a piece of metal weighing 
120 grams. The water that overflows weighs 15 grams. What is 
the specific gravity of the metal? 


VAN TUYL’S NEW COMP. AR.—13 


PRACTICAL MEASUREMENTS 

TIME 

444. The time between two dates may be expressed in three 
ways; viz. compound time, exact time, and bankers’ time. 

445. Compound time is expressed in years, months, and days. 
It is determined by the method of compound subtraction. The 
time from Dec. 14, 1921 to Oct. 11, 1924 is reckoned thus: 

1924 10 11 The com P oun d time is 2 yr. 9 mo. 27 da. Compound 

1921 12 14 time * S USe< * * n ma j° r ^y °f the ordinary business 

-- transactions involving time, especially if the period is 

2 9 27 more than a year. 

446. The method is based on a 360-da. year—12 mo. of 30 da. 
each. “One half a month” is fifteen days; one quarter of a year is 
three calendar months; and one half of a year is six calendar 
months. 

447. A month from a given day in any month is the same day in 
the next month except in those months in which there is no day to 
correspond with the given day in the month from which the time is 
reckoned. For instance, 1 mo. from Jan. 14 is Feb. 14, and 5 mo. 
from Feb. 1 is July 1. But 1 mo. from Jan. 29, 30, or 31 is Feb. 28 
(29th in leap year), and 3 mo. from Aug. 31 is Nov. 30, etc. 

448. There is a discrepancy of 5 da. per annum between com¬ 
pound time and exact time, but business men generally are willing 
to forego the difference on account of the facility of reckoning com¬ 
pound time. 

449. Exact time is expressed in days, or years and days. It is 
determined by counting the exact number of days between two 
dates. The exact time from May 3 to Sept. 28 is reckoned thus: 

May, 28 days remaining. 

June, 30 days. 

July, 31 days. Th e first date, May 3, is omitted; the last 

Aug., 31 days. date, Sept. 28, is included. For exceptions, 

Sept., 28 days. see Art. 685, note 1. 

148 days. 


194 



TIME 


195 


450. Exact time is used chiefly for periods of time less than one 
year, by the government in making interest calculations, by 
bankers in reckoning bank discount, and by some business men. 

451. Bankers’ time is expressed in days, or months and days. 
It is reckoned by counting months for the whole months and 
exact days for any remaining part of a month. To illustrate: 
From Jan. 20 to June 15, is 4 mo. and 26 da. 4 mo. from Jan. 20 
is May 20, and from May 20 to June 15 is 26 da. 

This method is used by bankers in reckoning interest on mort¬ 
gages, notes, etc., for fractional periods of time. 


PROBLEMS 


452. Find the compound time between: 

1. May 15, 1923 and Aug. 12, 1926. 

2. Apr. 4, 1920 and Jan. 3, 1926. 

3. Dec. 26, 1921 and Nov. 13, 1925. 

4. Oct. 13, 1923 and June 1, 1926. 

5. Sept. 1, 1921 and July 30, 1926. 

6. Sept. 28, 1924 and Feb. 14, 1927. 

7. Nov. 20, 1919 and Jan. 1, 1926. 

8. Dec. 31, 1922 and May 1, 1924. 

9. June 30, 1921 and May 15, 1924. 

10. July 4, 1920 and Dec. 25, 1924. 

Find the exact time and the bankers’ time between: 


11. May 4 and Oct. 9. 

12. April 24 and Dec. 1. 

13. Feb. 15 and Nov. 10. 

14. June 30 and Nov. 20. 

15. Jan. 15 and July 1. 


16. Feb. 18 and June 12. 

17. May 13 and Dec. 10. 

18. June 28 and Nov. 4. 

19. Apr. 16 and Sept. 5. 

20. Mar. 30 and Sept. 8. 


Find the compound time and the exact time between: 

21. Apr. 1, 1921 and Jan. 30, 1923. 

22. Mar. 30, 1922 and Feb. 28, 1923. 

23. Dec. 26, 1918 and Jan. 1, 1923. 

24. Nov. 20, 1922 and July 21, 1924. 

25. Oct. 18, 1921 and July 8, 1924. 

26. Sept, 17, 1922 and Aug. 12, 1923. 


196 


PRACTICAL MEASUREMENTS 


27. July 16, 1920 and June 1, 1923. 

28. June 14, 1922 and May 5, 1924. 

453 . This is a condensed time-table of the Atchison, Topeka 
and Santa Fe Railway, extending from Chicago to San Francisco. 



CHICAGO, KANSAS CITY AND CALIFORNIA 


WEST—Via La Junta and Albuquerque 


Mis 

STATIONS 

1 


The Scout 

9 

The Navajo 


Daily 


By Days 

Daily 

By Days 





PM 



AM 





0 

Lv Chicago.A.T.&S.F. 

8.55 

Su Mo Tu We Th Fr Sa 

9.25 

Su Mo Tu We Th Fr Sa 

451 

Ar Kansas City.. 

“ 

9.25 

Mo Tu We Th Fr Sa Su 

9.30 

a m u 

u 

u a 

u 

451 

Lv Kansas City. .A.T.&S.F. 

10.00 

Mo Tu We Th Fr Sa Su 

10.00 

Su Mo Tu We Th 1 r 

Sa 

578 

“ Emporia. 

U 

2.30 

U 

u u u u a u 

1.45 

Mo Tu We Th Fr Sa Su' 

684 

“ Hutchinson... 

u 

6.30 

u 

u u a u u u 

4.35 

u a u 

u 

u “ 

“ 

804 

“ Dodge City... 

a 

10.07 

u 

u u u u u u 

7.25 

u u u 

a 

a a 

“ 

1007 

“ La Junta. 

“ 

4.55 

Tu We Th Fr Sa Su Mo 

12.55 

u u u 

a 

a u 

a 

1222 

“ Las Vegas.. .. 

u 

2.00 

U 

u u u u u a 

7.55 

u u a 

u 

a a 

u 

1287 

Ar Lamy. 

“ 

4.55 

a 

a u a u u a 

10.40 

a a u 

a 

a u 

u 

1354 

Lv Albuquerque.. 

“ 

8.30 

a 

u a u u u a 

1.30 

Tu We Th Fr Sa Su Mo 

1515 

“ Galhip . 

u 

3.10 

We Th Fr Sa Su Mo Tu 

7.45 

u u a 

a 

a u 

u 

1702 

“ Flagstaff. 

u 

9.45 

“ 

u u u u u a 

1.40 

u a a 

u 

u u 

“ 

1786 

Ar Williams. 

u 

10.55 

U 

u u u u u a 

2.48 

u u a 

u 

u u 

“ 

1736 

Lv Williams. 

G. C. 

1.15 

We Th Fr Sa Su Mo Tu 



1800 

Ar Grand Canyon 


4.00 

“ 

a u u u a a 




Lv Grand Canyon 
Ar Williams. 

G. C. 

8.20 

We Th Fr Sa Su Mo Tu 

8.20 

Tu We Th Fr Sa Su Mn 



11.00 

“ 

a u u u u u 

11.00 

u a u 

“ 

a u 

“ 

1736 

Lv Williams.A.T.&S.F. 

11.05 

We Th Fr Sa Su Mo Tu 

2.55 

Tu We Th Fr Sa Su Mo 

1759 

Ar Ash Fork. 

a 

12.01 

U 

u a u u u u 

3.45 

u a u 

a 

u u 

u 

1936 

u Needles. 

a 

5.50 

U 

u u u u u u 

8.35 

u u u 

a 

a u 

u 

2105 

“ Barstow. 

u 

1.55 

Th Fr Sa Su Mo Tu We 

2.35 

We Th Fr Sa Su Mo Tu 

2186 

“ San Bernardino 

* 

5.30 

a 

u u u a u u 

5.35 

u u u 

“ 

a a 

“ 

2246 

“ Los Angeles... 

“ 

8.15 

Th Fr Sa Su Mo Tu We 

8.10 

a a a 

a 

a a 

“ 

2246 

Lv Los Angeles... 

u 

9.45 

“ 

u u u u a a 

9.45 

u u u 

a 

u u 

“ 

2373 

Ar San Diego.... 

“ 

1.20 

“ 

U U U U U u 

1.20 

u u u 

a 

u a 

u 

2105 

Lv Barstow.A.T.&S.F. 

3.00 

Th Fr Sa Su Mo Tu We 

3.00 

We Th Fr Sa Su Mo Tu 

2176 

“ Mojave. 

“ 

4.50 

“ 

u u u u u u 

4.50 

u u a 

“ 

u U 

a 

2245 

Ar Bakersfield. ... 

u 

7.45 

u 

u u u u u a 

7.45 

u u a 


u a 

u 

2414 

“ Merced . 

u 

12.55 

“ 

u u u u u u 

12.55 

u u u 

“ 

a u 

u 

2558 

“ Oakland . 

“ 

7.14 

“ 

“ “ “ “ a u 

7.14 

a a a 

a 

u u 

it 

2564 

Ar San Francisco. 


7A0 

u 


7.20 

a u a 

u 

u u 

u 




PM 



PM 










































































TIME 


197 


454. Use Central Time from Chicago to Dodge City; Mountain 
Time from Dodge City to Barstow, and Pacific Time from 
Barstow to San Francisco or San Diego. 

1. Find the average speed of train No. 1 from Chicago to 
Kansas City; from Kansas City to Albuquerque; from Chicago 
to Grand Canyon; from Kansas City to Los Angeles; from Albu¬ 
querque to Needles; from Needles to San Francisco; from Chicago 
to San Francisco. 

2. Find the running time of No. 9 from Chicago to Dodge City; 
from Chicago to La Junta; from Chicago to San Diego; from 
Chicago to Oakland; from Kansas City to Ash Fork; from Needles 
to Los Angeles; from Kansas City to San Francisco. 


455. Counting forward or backward from a given date. 

1. Count forward 76 da. from June 23. 


76 da 

7 da. remaining in June. 
69 

31 da. in July 
38 

31 da. in August 
7 September. 

Therefore, September 7. 


There are 7 da. left in June after June 23. 
Subtract 7 da. from 76 da., obtaining 69 da. 
to count forward after June 30. Subtract 
31 da. for July, which leaves 38 da. to count 
forward after July 31. Subtract 31 da. for 
August, which leaves 7 da. to count into 
September, or September 7. 


2. Count back 83 da. from May 10. 


83 da. 

10 da. back in May 

7$ 

30 da. in April 
43 


In counting backward from May 10, note 
that 10 da. are subtracted for May, leaving 73 
da. to count back from Apr. 30. Subtract 30 
da. for April, and 31 da. for March. The 12 
da. remaining are to be deducted from the 28 
da. in February, which leaves February 16. 


31 da. in March 
12 da. back into Feb. 
Feb. 28 - 12 da. = 
Feb. 16. 


Note. Fractional parts of a day are not 
recognized in law. Hence, as soon as it is past 
midnight on May 9, it is May 10, and the whole 
day is considered, in law, to be gone. There¬ 
fore, in counting back from the 10th, there are 


10 days to count back through to count back out of May. 


198 


PRACTICAL MEASUREMENTS 


PROBLEMS 

456. Count both forward and backward: 


1 . 

95 da. from July 10. 

ll. 

168 da. from Mar. 31,1922, 

2 . 

110 da. from 

June 30. 

12 . 

174 da. from Feb. 28, 1923, 

3 . 

57 da. from Apr. 15. 

13 . 

60 da. from July 1,1923. 

4 . 

132 da. from 

June 1 . 

14 . 

90 da. from May 1,1922. 

5 . 

160 da. from 

June 30. 

15 . 

89 da. from May 31,1922. 

6 . 

173 da. from 

Oct. 15, 1922. 

16 . 

101 da. from Nov. 1,1923. 

7 . 

208 da. from 

Sept. 1, 1922. 

17 . 

201 da. from July 31, 1923. 

8 . 

142 da. from 

Dec. 30, 1921. 

18 . 

179 da. from Aug. 19,1922. 

9 . 

111 da. from 

Jan. 10, 1922. 

19 . 

120 da. from Apr. 30, 1922. 

10 . 

131 aa. from 

Feb. 1, 1923. 

20 . 

229 da. from Dec. 1, 1923. 


PAINTING, PLASTERING, PAPERING, CARPETING 

457. Painting and plastering are estimated by the square yard. 

There is ho definite rule regarding allowances for openings. It 

is a matter for special agreement, and should be mentioned in the 
contract when made. 

458. Papering is estimated by the roll. 

459. Most American wall papers are 18 in. wide, a single roll 
being 24 ft. long, and a double roll, 48 ft. long. Imported papers 
vary in width and length. 

460. As there is always more or less waste in cutting and match¬ 
ing'wall paper, it is better to use double rolls. The exact number 
of rolls cannot always be determined in advance. 

461 . In estimating the number of rolls for a room, paper hangers 
generally deduct the total width of openings from the perimeter of 
the room. Then they reckon the number of strips to cover 
the remaining surface. The number of rolls is found by dividing 
the total number of strips required by the number of whole strips 
that can be cut from 1 roll. 

The spaces over and under the windows, over the doors, etc., are 
covered with the parts of strips left from each roll. 

462. Carpet is estimated and sold by the linear yard. Lino¬ 
leum is generally sold by the square yard. 


PAINTING, PLASTERING, PAPERING, CARPETING 199 


463. Carpets vary in width, the more common widths being 1 
yd. for Ingrain, and f yd. for Axminster, Brussels, Moquette, 
Velvet, and Wilton. 

464. Carpet may be bought in any length desired, but only in 
whole strips; hence, in order to determine the number of yards for 
a floor, the number of strips must be known.* The number of 
yards is then found by multiplying the yards in 1 strip by the 
number of strips. 

465. A room 16 ft. long, 14 ft. wide, and 8J ft. high, has 3 
windows each 3 ft. by 6 ft. and 1 door 3 ft. by 7 ft. Find the 
cost of plastering the walls and ceiling at $1.10 per square yard, 
allowing for the openings. 

2 X (16 ft. + 14 ft.) = 60 ft., perimeter of room. 

8| X 60 sq. ft. = 510 sq. ft. in the walls. 

16 X 14 sq. ft. = 224 sq. ft. in the ceiling. 

510 sq. ft. + 224 sq. ft. = 734 sq. ft. in ceiling and walls. 

3 X 3 X 6 sq. ft. = 54 sq. ft. in the windows. 

3 X 7 sq. ft. = 21 sq. ft. in the door. 

54 sq. ft. + 21 sq. ft. = 75 sq. ft. in all openings. 

734 sq. ft. — 75 sq. ft. = 659 sq. ft., net area of ceilings and walls. 

659 X $1.10 . , . 

---- = $80.54, cost of plastering. 

First find the perimeter of the room and then multiply the result by the 
height of the room. The result, 510 sq. ft., is the area of the walls. The 
ceiling is a rectangle, 16 ft. by 14 ft., whose area is 224 sq. ft. The total area 
of ceiling and walls is the sum of 510 sq. ft. and 224 sq. ft., or 734 sq. ft. Next 
find the total area of windows and doors and deduct the result from 734 sq. ft. 
to show the net area to be plastered. That gives 659 sq. ft. Dividing by 
9 sq. ft. to the square yard and multiplying $1.10 by the quotient gives 
$80.54, as the cost of plastering. 

* If any whole n*umber of strips exactly covered a floor, and there were no 
waste in matching the pattern, the number of yards for a floor could be found 
by dividing the area of the floor by the area of a yard of carpet, but these 
conditions occur so rarely that the statement above may be taken as correct. 



200 


PRACTICAL MEASUREMENTS 


466. In writing the dimensions of rooms it is customary to 
write the length first, then the width, then the height. It is 
also usual to write 10 ft. thus, 10', and 8 in., 8". Hence a room 

15 ft. long, 13 ft. wide, and 8 ft. 4 in. high, may be written “a room 
15' X 13' X 8' 4".” 

1. Find the cost of plastering the walls and ceiling of a room 
18' X 16' X 9', allowing for one half the area of 2 windows 
3' X 6' 6", and 2 doors 3' 6" X 7', at $1.20 per square yard. 

2. A room is 16' X 12' X 8' 6". It has 3 windows 4' X 5' 6", 
and 1 door 3' X 6' 6". At 95 a square yard, find cost of plaster¬ 
ing the walls and ceiling, making no allowance for openings. 

3. At 80 ^ a square yard find the cost of plastering the walls 
and ceiling of a room 26' X 16' X 10' 6", allowing for one half of 
the area of 4 windows 4' X 7' 6", and 3 doors 4' X 8'. 

4 . Find the cost of kalsomining the walls and ceiling of a room 
22' 6" by 18' by 9' 6", at $1.40 per square yard, making no allow¬ 
ance for openings. 

5 . Allowing 100 sq. ft. for openings, find the cost, at $1.05 per 
square yard, of plastering the walls and ceiling of a room 19' 6" X 
15' X 9'. 

How many yards of Brussels carpet are required for a room 

16 ft. long and 15 ft. wide? 

15 ft. -T- 2\ ft. = 6f = 7, no. strips. 

7 X 16 

-, = 37 J, no. yards. 

3 

Unless otherwise specified, carpet is generally considered to be laid with 
strips running lengthwise of the room. Hence, to find the number of strips, 
divide the width of the floor by the width of the carpet. 15 ft. 2\ = 6|, 
the number of strips, but as carpet has to be bought in whole strips, 7 strips 
are necessary. Each strip is 16 ft. long. The number of yards, therefore, is | 
of 7 times 16, or 37 

6 . How many yards of Wilton carpet are required for a room 
18 ft. by 16 ft.? 

7 . A hall 40 ft. long and 30 ft. wide has an ingrain carpet on the 
floor. Find the cost of the carpet at $2.50 per yard. 



PAINTING, PLASTERING, PAPERING, CARPETING 201 

8. A parlor is 16' 6" X 14'. How many yards of velvet carpet 
does it take for the floor? 

9 . A stairway has 12 steps each 10 inches wide. Each step has 
a riser of 8 inches. How many yards of carpet are required for 
the stairway? 

Find the number of rolls of paper required for a room 14 ft. 
by 12 ft. 6 in. wide, and 8 ft. 6 in. high, allowance being made 
for 2 windows each ft. wide, and for 2 doors each 3 ft. 9 in. wide. 

2 X (14 ft. + 12J ft.) = 53 ft., perimeter of room. 

(2 X 3J ft.) + (2 X 3J ft.) = 14§ ft., total width of opening^. 

53 ft. — 14J ft. = 38J ft., net perimeter to be papered. 

38| ft. -r- lj ft. = 25f = 26 strips required. 

48 ft. -T- 8| ft. = 5 whole strips from a double roll. 

26 strips -T- 5 strips = 5J = 6, no. rolls required. 

First find the perimeter of the room, and from it deduct the total width of 
all openings. That leaves 38§ ft. as the net perimeter to be papered. The 
number of strips of paper required is determined by dividing the net perimeter 
by the width of a roll of paper, hence 38§ ft. -J- 1§ ft- gives 25f, or 26 strips 
needed. Since paper is usually sold in double rolls, the number of strips that 
can be cut from 1 double roll is found by dividing 48 ft. by 8£ ft. (the length 
of 1 strip). That gives 5 strips from the double roll. Since 26 strips are 
needed, it takes as many double rolls as 5 strips are contained times in 26 strips, 
or 5 and a fraction, which makes it necessary to buy 6 double rolls. 

10 . Find the cost of papering a room 20' X 18' X 10'6"at$1.45 
a double roll; allow for 2 windows 4' wide; 1 door 3'6" wide. 

11. Find the cost of papering a room 18' 6" X 16' X 8' 6" at 95 £ 
a double roll; allow for 1 window 4' 6" wide; 2 doors 4' wide. 

12. Find the cost of papering a room 24' X 18' 6" X 10' at SI.30 
a double roll; allow for 3 windows 4' wide; 2 doors 4' 6" wide. 

13. Find the cost of papering a room 22' 6" X 19' X 9' 6" at 
SI-65 a double roll; allow for 2 windows 3' 6" wide; 2 doors 4' wide. 


202 


PRACTICAL MEASUREMENTS 


The diagram below is the first-floor plan of a modern farm¬ 
house. 

The rooms have a uniform height of 8' 6". The dining room and 
kitchen each have a wainscot 3' high. All others have a base 
board 9" wide. All windows are 5' 9" X 3". Outside doors are 
7' X 3' 4"; inside doors are 7' X 3' 2". The double doors between 
the sitting room and the parlor are 7' X 5' 8". (Measurements 
of windows and doors include casings.) 



14. How many square yards of linoleum will cover the floors of 
the kitchen and the pantry? 

15. Find the cost of ingrain carpet at $2.70 a yard for the dining 
room (carpet to be laid most economical way). 

16. Find the cost of papering the walls and the ceiling of the 
parlor at $1.25 per double roll. 




































ROOFING 


203 


17. How many yards of Brussels carpet will cover the floor of 
the sitting room? 

18. At $4.20 per yard, find the cost of Wilton carpet for the 
parlor. 

FLOORING 

468. Flooring is estimated by the square or by the thousand 
board feet. 


469. When lumber is “tongued” and “grooved,” or matched, as 
it is called, there is some waste, as lumber dealers always measure 
the lumber at its full width before it is 
matched. The amount of waste depends 
upon the width of the boards. Carpenters 
generally all^v one fifth for waste. That is, 
for 1000 sq. ft. of floor space, 1200 sq. ft. 

(board feet) of flooring is needed. 

470. How many square feet of flooring are required for the 
parlor represented on page 202? 



15' 2" = 15J';13'4" = 13J'* 
8 


91 40 w 0 728 

0 X 3 X S 3 ~ 


242f, 


Ans. 243 sq. ft. 


Reduce feet and inches to feet and 
multiply the length of the room by the 
width. The allowance for waste is best 
reckoned by multiplying by f. Using 
the cancellation form of solution, the 
result is readily obtained. 


PROBLEMS 

471. 1 . A floor is 18' 6” X 14' 4''. How many square feet of 
lumber are required to build it? 

2. Find the total amount of lumber required for all the floors in 
the diagram on page 202, except the woodshed. How much will 
it cost at $55 per 1000? 

3. Find the cost of a mosaic floor 60' X 32' if the tile cost $1.25 
per square foot, and the cost of laying is $8 per square. 

ROOFING 

472. Roofing is generally estimated by the square of 100 square 
feet. 

473. The most common roofing materials are shingles, slate, tin, 
asphalt, asbestos, and copper. 




204 


PRACTICAL MEASUREMENTS 


474. Shingles are estimated as having an average width of 4 in. 
and are laid 4 in., 4| in., 5 in., or 5§ in. to the weather, depending 
on the pitch of the roof. The steeper the pitch the greater the 
length exposed to the weather. The usual estimate per square is 
shown in the following table: 

475. A bundle contains 250 
shingles. A part of a bundle 
is not sold. Estimates, there¬ 
fore, have to be made in whole 
bundles. 

476. The pitch of a roof is 
numerically expressed by divid¬ 
ing the height ^ the span. 

When the height of the 


Table 


Length op Shingle 

Number op 

Exposed to the 

Shingles 

Weather 

Per Square 

4 inches 

1000 

\\ inches 

900 

5 inches 

800 

5| inches 

700 


ridgepole is J of the span of the roof above the building, 
the pitch of the roof is J. 

477. The accompanying diagram shows how the car¬ 
penter determines, or “lays off,” some of the more 
common pitches. 



PROBLEMS 


478. l. Using a span of 36 ft., make a diagram illustrating one 
third pitch; one fourth pitch; five twelfths pitch; five eighths 
pitch; equilateral or true pitch (60°). 

2. If the foot of the rafters projects 14” over the side of the 
building, find the length of the rafters for each of the pitches in 1. 

The rafter is the hypotenuse of a right triangle. 











PAVING 


205 


3. A roof is 60 ft. long and 30 ft. wide on each side. How many 
shingles laid 5 in. to the weather must be bought to cover it? 


2 X 60 X 30 
100 


= 36 squares. 


36 X 800 shingles = 28,800 shingles = 29,000 shingles. 

Each side of the roof is a rectangle 60 ft. by 30 ft. The area of both sides 
divided by 100 sq. ft. in a square, gives 36 squares. It takes 800 shingles for 
1 square, and for 36 squares it takes 36 times 800 shingles or 28,800 shingles, 
making it necessary to buy 29,000 shingles. 

4. A building is 36 ft. wide. The pitch of the roof is one half 
and the rafters project 15 in. How many shingles laid 4f in. to the 
weather are required if the ridgepole is 50 ft. long? 

5. How many slates 6 in. wide, 5 in. to the weather, are required 
for a building 24 ft. wide, the pitch of the roof being five eighths, 
the ridgepole 36 ft. long, and the rafters projecting 15 in.? 

6. A roof 44 ft. long and 18 ft. wide on each side is covered with 
tin. If each sheet is 18" X 27", how many sheets are there? 

7. How many slates are needed to cover a roof 32' X 16' on a side 
if the slates are 6" wide and 
are exposed 8" to the weather? 

8. The diagram shows part 
of the end of a barn having a 
gambrel roof. If the projection 
of the rafters is 18" and the 
ridgepole is 59 ft. long, how 
many shingles are required for 
the roof if they are laid 5| in. 

of the roof and 4 in. to the weather on the upper half? 



tn thpi wp.n.thp/r rm thp Inwpr bn If 


PAVING 

479. Paving is estimated by the square foot or by the square yard. 

PROBLEMS 

480. 1 . Find the cost of paving a street 8 rd. long and 30 ft. 
wide, at $1.75 per square foot. 

2. How many paving blocks 4" X 8" are required for a section 
of street 80 rd. long and 36 ft. wide? Find the cost of paving at 
$14 per square yard. 







206 


PRACTICAL MEASUREMENTS 


3. If paving stones are 5" X 9", how many will it take to pave a 
Street 1 mi. long and 40 ft. wide? 

4. A man who owns a corner lot with a frontage of 75 ft. and a 
depth of 100 ft., builds a cement walk 6 ft. wide on the side and 
front. If he puts the walk on this lot, find the cost at $3.75 per 
square yard. 

PAPER AND BOOKS 

481 . Paper is of various kinds, and has many uses, some of which 
are: 

1. Wrapping—made of straw, manila hemp, or wood pulp. It is used for 
wrapping bundles, making paper bags, etc. It is one of the cheapest kinds of 
paper. 

2. News—made of wood pulp. It is of a higher grade than wrapping paper, 
and is used in making newspapers, and cheap books and magazines. It is sold 
chiefly in large rolls. 

3. Book—made of wood pulp and cotton cloth (called “paper rags”). It 
is better than news and is used in making the better class of books and maga¬ 
zines. It is sold chiefly in large unfolded sheets; also in large rolls like news. 

4. Flat—made of wood pulp and cotton or linen cloth, mixed with an 
animal size, or glue. The sizing produces a firm smooth surface and prevents 
the ink from spreading in the paper when written upon with a pen. It is used 
for correspondence, and for records and documents in which permanency is 
desired. It is sold wholesale in large unfolded sheets. For retail purposes it is 
cut into various-sized sheets dependent upon the use to which it is to be put. 

482 . Paper is sold in large quantities by weight; in small quan¬ 
tities, by the quire or ream. In some cases a ream consists of 500 
sheets instead of 480. 




Common Sizes of Paper 



Book 

Flat 



25” 

X 38” 

Cap. 

. 14” 

X 17” 

28” 

X 42” 

Demy. 

. 16” 

X 21” 

30|” 

X 41” 

Folio. 

. 17” 

X 22” 

32” 

*X 44” 

Medium .... 

. 18” 

X 23” 

35” 

X 46” 

Super Royal . . 

. 20” 

X 28” 


The 25” X 38” size is the basis from which the weights of other sizes of book 
paper are reckoned. A ream 25” X 38” weighs 40 lb., 50 lb., 60 lb., etc., ac¬ 
cording to the quality of the paper. 

484. Formerly the terms folio, quarto, 8vo, 12mo, etc., in¬ 
dicated the number of times a sheet was folded in making a book. 
For instance, folio was the name given to a book made of sheets 





PAPER AND BOOKS 


207 


which had been folded but once, thus making two leaves or four 
pages from each sheet; a quarto indicated a book made of sheets 
folded twice, making four leaves or eight pages; etc. 

At the present time the terms folio, quarto, octavo, or 8vo, have 
reference to the size of the page, regardless of the number of times 
a sheet is folded. 

485. The following sizes are the commercial standard board 
(cover) measurements of various kinds of books: 

Quarto.12f" X 8f 16mo.6|" X 

8vo.9|" X 6" 18mo.6i" X 3f" 

12mo . 7f" X 4f" 32mo.4f" X 3i" 

In practice any of the above sizes may run large or small; hence 
it is impoossible to tell the exact size of a book by the terms folio, 
quarto, octavo, etc. 

486. Most books, except very large ones, are printed in 16’s or 
32’s; that is, each sheet has 16 or 32 pages on each side. The 
sheets are then folded and cut into two or four sections called 
“signatures.” Each signature contains 16 pages. Hence a sheet 
is folded to make either 32 or 64 pages. 

487. 1 . If a sheet is folded into 4 signatures, each page 7|” X 
4£”, which of the above sizes of paper is it best to use? 

4 signatures = 64 pages = 32 pages on one side of sheet. 

32 pages = 8 pages by 4 pages. 

4 X 7J” = 30”, width of sheet. 

8 X 4f ” = 39”, length of sheet. 

Hence the size 30J” X 41” is best. 

2. A book containing 448 pages is printed in 32 , s. If the pages 
are 7J” X 5f ”, what size paper would be used? How many reams 
(500 sheets) would be required to print 5000 copies of the book? 

32’s means 32 pages on one side of a sheet. 

32 pages = 8 pages by 4 pages. 

4 x 7J” = 31”, width of sheet. 

8 X 5J” = 42”, length of sheet. 

Hence, use sheets 32” X 44”. 

448 pages - 4 - 64 pages (on 1 sheet) = 7, no. sheets in 1 book* 

5000 X 7 ™ 

-—— = 70, no. reams. 

500 









208 


PRACTICAL MEASUREMENTS 


3. If 70 sheets of strawboard 26" X 38" weigh 501b., what will 
be the cost of 1000 sheets of the same grade of strawboard 8J" X 17" 
at 828 per ton? 

First cause: Second cause:: First effect: Second effect. 

70 sheets : 1000 sheets 

26 in. : 8f in. :: 50 lb. :? lb. 

38 in. : 17 in. 


1000 X 8J X 17 X 50 


lb. = 104J lb. 


By the “cause and effect” 
method of stating the propor¬ 
tion, it is readily seen that the 
first cause is the 70 sheets 26” 
X 38”, and that the 


corre- 


70 X 26 X 38 

$28 -5- 2000 = $.014 per pound. 

104J X $.014 = $1.46, cost, 
sponding effect is 50 lb. The second cause is the 1000 sheets 8|" X 17", leav¬ 
ing the second effect to be found. Dividing the product of the means by the 
given extreme gives the other extreme, or 104§ lb., the weight of 1000 sheets 
X 17 ;/ . Multiplying by the price per pound gives $1.46, the required cost. 

With a little practice the student should be able to write the above state¬ 
ment directly in the form for solution, thus: 

50 X 8f X 17 X 1000 X $28 


70 X 26 X 38 X 2000 


= $1.46 


reasoning in this way: If 70 sheets 26” X 38” weigh 50 lb., one sheet will 
weigh T V of 50 lb., and 1 sq. in. of 1 sheet will weigh as much as 26 X 38 
is contained times in the weight of one sheet. Thus far the statement would 


50 


be —--—-— = weight of 1 sq. in. If 1 sq. in. of 1 sheet weighs that much, 

70 X 26 X 38 

one sheet X 17” will weigh 8£ X 17 times that, and 1000 sheets will weigh. 

50 X Sh X 17 X 1000 


1000 times that result. To this point the statement is 


70 X 26 X 38 


Having the weight, divide by 2000 lb. to a ton, and multiply by the price per 
ton. The complete statement is as given above. 


PROBLEMS 

488. l. If two signatures are printed from one sheet, what size 
paper is best for a book whose pages are Ilf" X 8f"? 

2. A 24-signature book is printed in 32’s. If the pages are 6f" 
X 5", what size of sheet would be best to use? How many 
volumes could be printed from 100 reams (480 sheets) of paper? 

3. If 1 ream (500 sheets) of book paper 26" X 38" weighs 50 lb., 
find the cost of 70 reams of the same grade 28" X 42" at $85 per 
ton. 






LUMBER 


209 


4. A ton of flat paper 17" X 22" costs $.15 a pound and is made 
into 10 notebooks 5|" X 8J", of 64 pages each. If 12 sheets 
of paper weigh 1 lb., and the cost of manufacturing the books in¬ 
cluding the covers is $1 per hundred, find the profit. 

5. If a ream of Demy weighs 50 lb., and a firm is charged for 
1075 lb. at 22 per pound for 80 reams 8" X 10J", how much too 
great is the charge? (A discrepancy of 1 lb. in 20 lb., or 5%, is not 
considered an overcharge in the paper trade.) 

The regulation size of letter paper is 8" X 10"; of note paper is 
5" X 8". (Commercial note is 5J" X 8J".) 

6. How many reams of paper, letter size, can be cut from 2000 
sheets 16" X 21"? 

7. If 24 reams 17" X 22" (500 sheets to a ream) are cut into 
commercial note size and sold at $1.10 per ream (480 sheets to a 
ream), what is the amount received? 

8. How many pads, 80 sheets each, 5f" X 9", can be made from 
4000 sheets 18" X 23"? 

9. How many reams (500 sheets to a ream) Super Royal are re¬ 
quired for 500 ledgers of 320 pages each, if the page is 10" X 14"? 

10 . A publisher has an order for 10,000 copies of a book contain¬ 
ing 512 pages, each page 8|" X 5|". He prints in 32’s, and the 
paper costs $100 per ton. If the same grade of paper 25" X 38" 
weighs 50 lb. to the ream, find the cost of the paper for the order. 

LUMBER 

489. The unit of lumber measure is the board foot. A board 
foot is a square foot of board one inch (or less) thick. Each of 
these illustrations represents a board foot of lumber, 
sjo/- 7 ) 1 in - „ 

v. _/ # —— . ^ 

1 ft. 2 ft. 4 ft. 

" f -if 1 ”' < £ J * 1 *- ' 

2 ft. 3 ft. 

The volume of a board foot of lumber is thus seen to be 144 cu. 
in., except when the lumber is less than 1 in. thick, in which case a 
board foot is equal to a square foot of surface of the board. 

VAN TUYL’S NEW COMP. AR.—14 

















I 


210 PRACTICAL MEASUREMENTS 

490. How many board feet are there in 48 scantlings 2" X 4" 
X 16' (2 in. X 4 in. X 16 ft. long)? 

4 

X 16 X 2 X 4 _ 10 ,, +r 

-= 512, no. of board feet in 48 scantlings. 

Xa 

When there is more than one piece of lumber, the simplest method is to 
find the total length of all the pieces, and then multiply by the width and by 
the thickness, in inches, and divide the product by 12, using cancellation, as 
shown in the solution. 


LUMBER YARD PRACTICE 

491. To find the number of board feet in one piece of lumber. 

4 inches wide, take | of the length. 

6 inches wide, take § of the length. 

8 inches wide, take | less than the length. 

9 inches wide, take \ less than the length. 

10 inches wide, take \ less than the length. 

12 inches wide, take the length. 

14 inches wide, add £ to the length. 

15 inches wide, add \ to the length. 

2 in. by 4 in. (2 X 4 = 8) take | from the length. 

2 in. by 8 in. (2 X 8 = 16) add 1 to the length. 

8 in. by 8 in. (8 X 8 = 64) take 51 times the length, etc. 

Note. When no thickness is mentioned, lumber is understood to be one 

inch thick. 

PROBLEMS 


492. Find, mentally when possible, the number of board feet in 
the following: 


1. 

20 

pcs. 

3" 

X 

8" 

X 

14' 

10. 

200 

pcs. 

4" 

X 

8" 

X 

10' 

2. 

44 

pcs. 

4" 

X 

4" 

X 

18' 

li. 

14 

pcs. 

3" 

X 

5" 

X 

14' 

3. 

60 

pcs. 

12" 

X 

1" 

X 

16' 

12. 

18 

pcs. 

4" 

X 

9" 

X 

16' 

4. 

90 

pcs. 

2" 

X 

4" 

X 

16' 

13. 

28 

pcs. 

6" 

x: 

12" 

X 

18' 

5. 

160 

pcs. 

6" 

X 

8" 

X 

12' 

14. 

38 

pcs. 

12" 

x : 

12" 

X 

16' 

6. 

16 

pcs. 

10" 

X 

12" 

X 

18' 

15. 

560 

pcs. 

10" 

X 

1" 

X 

16' 

7. 

120 

pcs. 

9" 

X 

11" 

X 

16' 

16. 

960 

pcs. 

8" 

X 

1". 

X 

14' 

8. 

75 

pcs. 

4" 

X 

6" 

X 

14' 

17. 

760 

pcs. 

6" 

X 

1" 

X 

18' 

9. 

49 

pcs. 

8" 

X 

8" 

X 

22' 

18. 

1000 

pcs. 

5" 

X 

1" 

X 

16' 



PRACTICAL MEASUREMENTS 


211 


493 . This diagram shows 
the framework of the ground 
floor plan of a house 34'X51'. 
The sills are 6" X 8". The 
side sills are spliced 5 ft. in 
the middle. The joists are 
3" X 8", and are 18" apart 
from center to center. Find 
the cost of the lumber at $38 
per M. 


3 sills 

4 sills 


6" X 8" 
6" X 8" 


42 joists 3" X 8" 
Total, 


X 34' = 408 ft. 
X 28' = 448 ft. 
X 26' = 2184 ft. 


3040 ft. 
3.040 X $38 = $115.52, cost. 


51 feet 

The three cross sills are each 
34 ft. long, making 408 board 
feet. The side dimensions are 
so long as to make splicing 
necessary. The length of each 
piece of timber is found by 
adding the length of the splice 
to the length of the sill and 
dividing into two equal parts. 
(5 + 51) 2 = 28, no. ft. 

. The four pieces contain 448 

board feet. 

Since the sills are 6" thick, the width inside the sills is 33'. To find the num¬ 
ber of joists divide 33' by 18", and subtract 1 Rom the quotient. 33' -f- 18" = 

22;22 — 1 = 21. (The number of spaces is 22, 
but the number of joists is one less. Prove it 
by counting spaces and joists in the diagram.) 
As one joist reaches only to the middle sill, 
there are 2 times 21, or 42, joists in all. They 
are mortised into the sills 2" at each end. The 
total width of the three cross sills is 18", which, deducted from 51', leaves 
49'6", net length of house exclusive of the 
sills. One half of 49' 6 ' equals 24' 9", length 


of joist exclusive of the tenon, — 2" at each 
end,— which makes total length of a joist H0W ^ sfucu, 

25'1". (24'9" + 2" + 2".) But as lumber is sawed in even foot lengths, a 

joist 26' long must be bought. 42 joists 3" X 8" X 26' = 2148 board feet. 

The total amount of lumber is 3040 ft., which at $38 per M is worth $115.52 



Note. Lumber is practically never cut in an odd number of feet lengths, 
as 13', 15', 19', 25', but in even numbers, as 10', 12', 16', 22', etc. Hence, 
when the actual length required in building is an odd or fractional number, the 
nearest length in even number of feet greater is purchased. 































212 


PRACTICAL MEASUREMENTS 


PROBLEMS 

494. l. Estimate the cost of the lumber at $44 per M for a simi¬ 
lar plan 25' 4" X 40'. The sills are 8" X 8", those on the sides 
being spliced 4 ft. The joists are 3" X 8" and are 16" from center 
to center. 

2. A house is 25 ft. by 30 ft. The sills are 6" X 6". A sill ex¬ 
tends across the middle of the plan as in the preceding diagram. 
The joists are 3" X 6" and are 16" from center to center. How 
many feet of lumber are required for the sills and joists? 

3. Make a diagram of an original plan similar to those of the pre¬ 
ceding problems, and estimate the amount of lumber necessary to 
construct. 

WOOD 

495. Wood is measured by the cord of 128 cu. ft. A pile of 
wood, 8 ft. long, 4 ft. wide, and 4 ft. high is a cord. 1 ft. of the 
length of such a pile is sometimes called a cord foot. 

PROBLEMS 

496. l. A pile of wood is 44 ft. long, 24 ft. wide, and 12 ft. high. 
How many cords are there? 

2. A woodshed is 16 ft. long, 12 ft. wide, and 12 ft. high. How 
many cords of wood can be piled in it? 

3. A dealer bought a pile of wood 120 ft. long, 6 ft. high, and 36 
ft. wide at $5.75 per cord. He hired it sawed into stove lengths 
(16 in.) at $1.75 per cord. He sold it at $9.50 a cord, paying 
75 i a cord for delivering. Find his gain. 

4. A wagon rack is 18 ft. long, 3 ft. 3 in. wide. How high must 
the wood be piled to make a load of 2 cd.? 

5. How many steres of wood are there in a pile 48 ft. long, 24 ft. 
wide, and 9 ft. high? (1 stere = 1 cu. m.) 

6. A woodshed is 20 ft. long, 8 ft. high on one side and 12 ft. 
high on the other side, and is 16 ft. wide. Allowing space about 
the entrance for three cords, find how many cords of wood can be 
piled in it. 

7. A freight car is 33' X 7' X 8' inside measurement. How 
many cords of wood can be piled in it? 


APPROXIMATE MEASURES 


213 


CAPACITY 

BINS AND GRAIN ELEVATORS 

497. A bin is a box or inclosed space for holding grain, coal, 
fruit, vegetables, etc. 

498. A grain elevator is a building for the storage of grain. 

499. Grain and vegetables (especially potatoes) are sold by the 
bushel. The number of bushels is determined by weight rather 
than by volume. Estimates of the number of bushels a box or bin 
will hold are made in bushels of capacity. 

500. Estimates of grain are made by the stricken bushel of 
2150.42 cu. in., fruits and vegetables by the heaped bushel, 
2747.71 cu. in. 

Fruits are sold in barrels, boxes, and baskets of various sizes. 

501 . Coal is sold by weight. It is sold wholesale at the mines by 
the long ton; it is usually retailed by the short ton. In estimating 
the capacity of a bin, 34| cu. ft. of space are reckoned to contain 
1 ton of hard stove coal. 


APPROXIMATE MEASURES 


502. Since a bushel of grain measures 2150.42 cu. in., and there 
are 1728 cu. in. in a cubie foot, a cubic foot of space contains 


-, or .8 + bu. Hence, 

2150.42 

To find the number of bushels of grain a bin will hold, multiply its 
capacity measured in cubic feet by .8. 

This rule is sufficiently accurate for all practical purposes, the 
error being less than 4 bu. per thousand cubic feet. 


In like manner, 


a 


cubic foot of space contains 


1728 

2747.71 


, or .63 bu., 


of potatoes, apples, etc., and to find the number of bushels that can 
be put into a box or bin, multiply its capacity in cubic feet by .63. 
The error is less than 1^ bu. per thousand cubic feet. 

Use approximate measures, unless otherwise specified. 

503. How many bushels of wheat can be put into a bin 20' X 

8' X 67 20 X 8 X 6 X .8 bu. = 768 bu. 

Since a cubic foot of space contains .8 of a bushel of wheat, a bin 20' X 8' 
X 6' contains 20 X 8 X 6 X .8 bu., or 768 bu. 




214 


PRACTICAL MEASUREMENTS 


PROBLEMS 


504. l. Estimate the quantity of grain that can be stored in a 
grain elevator 30' X 40' X 28'. 

Estimate the quantity of (a) grain, (6) potatoes, that can be put 


into each of the following bins 

2. 8' X 6' X 4' 

3. 9' X 8' X 10' 

4. 4' X 5' X 20' 

5. 12' X 8' X 36' 

6. 15' X 9' X 40' 

7. 10' X 5' X 4' 

8. 9' X 10' X 12' 

16. How long a bin 6 ft. wide, 
bu. of oats? 


9. 4' 6" X 5' X 8' 

10. 5' 4" X 7' X 9' 

11. 3' 6" X 8' X 10' 6" 

12. 4' 4" X 8' 3" X 12' 

13. 8' 6" X 9' X 10' 

14. 7' 6" X 9' X 12' 

15. 2' 6" X 3' 3" X 4' 

4 ft. deep, is necessary to hold 480 


17. A bin is 36 ft. long and 5 ft. wide. To what depth will 756 
bu. of potatoes fill it? 

18. If a bin holds 1000 bu. of grain, how many bushels of potatoes 
will it hold? 


19. How many tons of stove coal can be put into a bin 8' X 
8' X 6'? 

20-24. Find the exact capacities for both grain and potatoes of 
the bins mentioned in problems 3-7. 

25. A farmer sells 8 loads of potatoes whose gross weights are 
4229 lb., 4301 lb., 4408 lb., 4027 lb., 4187 lb., 4360 lb., 4110 lb., and 
4568 lb. If the wagon w T eighs 1073 lb., how much does he receive 
for his potatoes at $.95 per bushel? 

26. A carload of wheat weighs 45,000 lb. If the car is 33 ft. long, 
7 ft. wide, and is filled to a depth of 4 ft., how much does the meas¬ 
ure by weight differ from the measure by capacity? 

27-33. How many hectoliters of grain will each of the bins in 
problems 3-9 hold? 


CISTERNS AND RESERVOIRS 

505. Cisterns and reservoirs are storage places for water, oil, etc. 
The unit of measure is the gallon of 231 cu. in. One cubic foot of 
space contains -Y^tS or 7.48 + gal. For mental work, 7J gal. are 
reckoned for each cubic foot. 


APPROXIMATE MEASURES 


215 


At 7.48 gal. to the cubic foot, the result varies from the result by 
exact measurement by less than 1 gal. for each thousand cubic feet 
of space. 

To find the number of gallons in a cistern or reservoir , multiply the 
capacity in cubic feet by 7.48. 

506 . A reservoir is 44' X 35' X 12'. Find (a) the exact number 
of gallons, and ( b ) the approximate number of gallons it will hold. 


(a) 


44 X 35 X 12 X 1728 
231 


138,240, no. gal. 


(6) 44 X 35 X 12 X 7.48 =138,230.4, no. gal. 

(a) To find the exact number of gallons, reduce the cubic feet to cubic 
inches and divide by 231 cu. in. in a gallon. The result is 138,240 gal. 

( b ) To find the approximate capacity, multiply the number of cubic feet 
by 7.48. The result is 138,230.4 gal. 


PROBLEMS 


507 . Find (a) the exact, and (6) the approximate, number of gal¬ 
lons in each of the following cisterns and reservoirs: 


1. 8' X 8' X 8' 

2. 9' X 6' X 7' 

3. 46' X 25' X 10' 

4. 55' X 49' X 18' 

5. 6' X 7' X 6' 


6. 5' X 5' X 10' 

7. 15' X 12' X 6' 6" 

8. 8'3" X 10' X 12'6' 

9. 9' 6" X 12' X 15' 

10. 12' X 12' X 12' 


11-16. Find approximately (a) the capacity in liters, (5) the 
weight in kilograms, of pure water in the cisterns and reservoirs in 
problems 1-6, supposing each is full of water. 

17. How many gallons will a cylindrical tank 24 ft. in diameter 
and 16 ft. deep hold? 

24 X 24 X .7854 X 16 X 7.48 = no. of gallons. 

24 X 24 X 16 X 5J = 54,144 gal. 

The volume of a cylinder is found by multiplying the square of the diam¬ 
eter by .7854 and multiplying that product by the altitude. To find the 
number of gallons, multiply the result thus found by 7.48. 

This process results in a long and tedious multiplication. Note that the 
second statement in the solution differs from the first in that the number 
.7854 and 7.48 are omitted, and 5| is used in their stead. 5| is the product 
of 7.48 X .7854, very nearly. Therefore the capacity of a cylindrical cistern 
or reservoir may be found (very nearly exactly) by multiplying the square 
of the diameter times the depth by 5|. 



216 


PRACTICAL MEASUREMENTS 


18. A cylindrical cistern is 10' in diameter and 10' deep. Find 
by the method just explained how many gallons it will hold. How 
much does the answer vary from the exact capacity? 

Find the approximate capacity in gallons of tanks having 
measurements as follows: 

19. Diameter 4', depth 8'. 22. Diameter 18', depth 20'. 

20. Diameter 5', depth 10'. 23. Diameter 44', depth 42'. 

21. Diameter 55', depth 56'. 24. Diameter 16', depth 10'. 

25. If a standpipe 235 ft. high holds 22,090 gal. of water, what is 

its diameter? 

26. A cylindrical reservoir 48 ft. in diameter holds 649,728 gal. 
What is its depth? 

LAND 

508. Land areas are estimated by the acre of 160 sq. rd. 

509. In laying out public lands, surveyors select a north and south 
line as a principal meridian, and an east and west line as a base line. 



6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 


Townships, 6 miles square A Township divided into Sections 

Other lines 6 mi. apart are run parallel to the principal meridian 
and to the base line, thus dividing the land into townships, as 
shown in the diagram. 

Rows of townships north and south are called ranges, and are 
numbered both east and west from the principal meridian. The 
townships are also numbered north and south of the base line. 
Thus, Township x is 5 north, in the 4th range east of the principal 
meridian. 




























LAND 


217 


Each township is again divided into squares by parallel lines 1 
mi. apart. Each square is called a section, and 
contains 640 A. 

The sections in a township are always num¬ 
bered as shown in the diagram. If the diagram 
of a township on p. 216 represents Township x 
of the preceding diagram, section 17 would be 
designated as section 17 of township 5 north, in 
the 4th range east of the principal meridian. 

Sections are divided into halves, quarters, half-quarters, quarter- 
quarters, etc. 


N.W. % 
of 

\.w. h 


E.J* 

Section 

320 A. 

S. Kof 
N.W. % 

S.W.J 4 
Section 

160 A. 


A Section 


510. 


PROBLEMS 

1. A man owned the southeast quarter of a section of land. 


He bought the south half of the northeast quarter, and the north¬ 
east quarter of the northeast quarter of the same section. Make a 
diagram of his original farm, adding thereto the areas bought, 
showing number of acres in each parcel. 

2. A man owning the north half of a section bought the east 
half of the southeast quarter, and the west half of the southwest 
quarter of the same section. How many acres 
of land has he? Draw a diagram of his farm. 

3. A railroad extends through a section of 
land as shown in the diagram. It occupies a 
strip 4 rd. wide. Find the land damage for the 
right of way at $40 an acre. 

4. Find the cost at $.65 a rod of building a 
fence around a section of land; a quarter section 
of land. 

5. Your deed shows that you own the southerly half of section 
8 in Clark Co., S. Dak. Draw a township plan, and locate your 
land. Locate a house in the southeastern corner of your land. 
Locate a haystack in the northwestern corner of your land. Find 
distance in a direct line from the house to the haystack. 

6. How many rods of fence are required to inclose each of the 
three parcels in problem 1? 

7. How many rods of fence will inclose the combined tracts men¬ 
tioned in problem 2? 









218 


PRACTICAL MEASUREMENTS 


511 . An estimate is an approximate result obtained by making a 
rough calculation of a given problem. Estimates are often made 
by inspection, and in advance of careful, accurate solution. 

1. Estimate the quantity of lumber required to build a porch 8 
ft. wide along the front of a house 25 ft. wide. The flooring is of 
matched lumber and is laid on joists 2" X 6". The joists are 17 
in number and are supported by sills 8" X 8". The porch is 
ceiled with matched lumber. The ceiling is nailed to 11 joists 
each 2" X 4". There are 13 rafters 2" X 4". The roof boards 
are 6" wide. 

2. Estimate the quantity of lumber required to build shelves on 
one side of a room 40 ft. long, if there are 7 shelves one above the 
other 14" apart, and 12" deep. The shelves are supported by 
board partitions 30" apart. 

3. Reckon the cost of painting (2 coats) a house 28' X 36', the 
eaves being 18 ft. from the top of the cellar wall. Use local prices 
of paint and labor. One gallon of paint will cover 400 sq. ft. first 
coat, and 500 sq. ft. second coat. One man will spread 3 gal. of 
paint first coat and 2 gal. second coat. Add 15% for contractor’s 
profit. 

4. Make a list of articles necessary for furnishing a 5-room 
apartment. Estimate the cost for each room, the total cost not to 
exceed $1000. 

5. Assume that your house has been destroyed by fire. Make 
an insurance inventory of the articles burned with the values 
thereof, as a basis of settlement with the insurance company. 

6. A quarter section of land in the form of a square is to be 
divided into 4 fields of 40 acres each, in such manner as to require 
the least amount of fencing. Make a diagram of the quarter sec¬ 
tion showing the position of the fences. Estimate the quantity of 
lumber required for a board fence inclosing the four fields with 6 
in. boards, if the fence is 4 boards high. If the posts are 8 ft. 
apart, how many posts are needed? 

7. A man owns a city lot 100 ft. deep and 75 ft. front. He de¬ 
sires to build a picket fence on the two sides and across the back 
end of the lot. Consult local builders and lumbermen for prices 
and materials, and estimate the cost of the fence. 


FARM PROBLEMS 


219 


8. Compare the cost of the picket fence in problem 7 with 
the cost of a moderately priced iron fence. Consult a hardware 
dealer for prices and lasting qualities, and determine which is the 
more economical, the picket or the iron fence. 

9. Investigate and find the cost of a thorn hedge across the 
front of the lot mentioned in problem 7. Make allowance for 
gateways for a walk to the house and for a driveway leading to the 
garage. 

10. Consult local contractors and determine the relative costs of 
covering a roof 36 ft. long having 20 ft. rafters, with #1 cedar 
shingles and with slate. Which is the more economical, considering 
durability? 

11. A man desires to inclose his front porch with portable wire 
screens for summer use, and with portable glass sash for use as a 
sun parlor in winter. If the porch is 28 ft. long and 12 ft. wide, 
estimate the cost of the screens and sash. 

12. It is estimated that there are 13,000,000 automobiles in the 
United States. Assuming that each automobile requires, on the 
average, four new tires a year, and that the average price of tires is 
$35, what is the approximate value of the tire industry in the 
United States per annum? 

FARM PROBLEMS 

512. l. A field of potatoes is one quarter of a mile long. If the 
rows are 3 ft. 8 in. apart, how many rows make an acre? 

2. If the field in problem 1 yields 10 bu. to the row, how many 
bushels will there be from 10 acres? 

3. If potatoes shrink 1% a month for five months, what will be 
the loss in weight on the above crop of potatoes from October to 
March? 

4. If the price of potatoes in October was 70 £ and in March it 
was 80 £, did it pay to hold the crop till March before marketing 
How much was gained or lost by holding the crop? 


220 


PRACTICAL MEASUREMENTS 


5. During the month of June one cow gives an average of 3 gal. 
of milk a day which tests 3.6% of butter fat; another cow gives 
an average of 2| gal. of milk testing 4% butter fat. Milk weighs 
8| lb. per gallon. If butter is worth 48^ a pound, find the value of 
the butter produced by each of the. cows. 

6 Complete the following ticket: 


Ticket Showing Weights of Hay 


Gross Weight 

Weight of Wagon 

Net Weight 

Value at $16.50 a Ton 

3674 

1467 

— 

— 

3860 

1490 

— 

— 

4135 

1505 

— 

— 

Totals 





7. Prepare a ticket like the above for 6 loads of coal weighing as 
follows: Gross weights, 4530,4654,4850,5260,4960, 5210; Weights 
of wagons: 1650, 1642, 1528, 1470, 1465, 1640. Coal is worth 
$10.50 per ton. 

8. A farmer’s corn field is 40 rd. long. If the rows are 3 ft. 8 in. 
apart each way, how many rows lengthwise of the field make an 
acre? How many hills of corn are there to the acre? 

9. Compare two crops from the field in problem 8, under the fol¬ 
lowing condition: (1st) 3 stalks to the hill, each stalk bearing one 
good ear of corn, 100 ears making a bushel, and the corn selling for 
70 i a bushel; (2d) 5 stalks to the hill, each stalk bearing a small 
ear of corn, 160 ears being required for a bushel, and the corn selling 
at 65 i a bushel. Which is the better crop, and how much better 
per acre? 

10. Estimate the number and size of pieces of lumber required to 
construct a corncrib 24 ft. long, 5 ft. wide at the bottom, 8 ft. 
wide at the top, and 6 ft. high to the foot of the rafters. The ridge 
of the roof rises 3 ft., and the rafters project 1 ft. Also estimate 
the number of board feet of lumber, and the number of shingles 
required. 











GRAPHS 


221 


GRAPHS 

513. A graph is a diagram illustrating some relationship. The 
relationships illustrated by graphs are of several kinds. One 
kind is the relation existing among a series of values of the same 
kind occurring in chronological order, such as the monthly or 
annual income or disbursements of a business organization. Such 
a graph is spoken of as a “curve,” and shows not only the relation 
of successive values to one another but also the “trend” of the 
series of values— i.e., whether the tendency is for them to increase 
or decrease. 

Another kind of graph shows the relation existing among each 
of several values and also of each value to the sum of all the 
values. A circle, bar or other form, used to represent the total, 
is used. 

A third kind of graph is used to show the relation of several 
values to one another. Such a graph is called a “bar” graph. 

General relationships are often illustrated by means of pictures 
showing comparative sizes, numbers, values, etc. 

The following two graphs show the relation and the trend of 
the earnings and expenses of a railroad (1) for a series of years, 
and (2) each month for one year. 



1916 


1920 


1921 




















































































































222 


PRACTICAL MEASUREMENTS 


From 1916 to 1921 inclusive, 
the annual earnings and ex¬ 
penses of the X Railway were 
as follows: 

The monthly earnings and 
expenses of the same railway 
for 1921 were as follows: 


Expenses 

Earnings 


Earnings 

Expenses 

1916 

$460,000 

$ 660,000 

January . . 

$ 78,000 

$63,000 

1917 

470,000 

670,000 

February . 

73,000 

61,000 

1918 

485,000 

800,000 

March . . 

87,000 

63,000 

1919 

640,000 

960,000 

April . . . 

86,000 

61,000 

1920 

680,000 

1,000,000 

May . . . 

95,000 

70,000 

1921 

850,000 

1,120,000 

June . . . 

101,000 

73,000 




July ... 

115,000 

82,000 




August . . 

119,000 

87,000 




September . 

98,500 

84,000 




October . . 

92,000 

86,000 




November . 

85,500 

70,000 




1 December . 

79,000 

65,000 



For a recent year the Congress of the United States appro¬ 
priated sums of money for the promotion of vocational education 
in the several groups of states as follows: 1. North Atlantic group 
$935,000. 2. Southern group $690,000. 3. East Central group 
$930,000. 4. West Central group $280,000. 5. Pacific group 
$215,000. 


























































































































































GRAPHS 


2 23 


Graphically these results 
would appear as here illus¬ 
trated. 

At one time the number 
of disabled soldiers, sailors 
and marines pursuing cer¬ 
tain vocational courses in 
Business and Commercial 
Training offered by the 
Government was as follows: 

Administrative positions 

. 2763 

Subordinate positions 
Commercial facilities 



The bar graph showing these figures is as follows: 


Administrative 

positions 

Subordinate 

positions 

Commercial 

facilities 



3000 


l. Illustrate graphically: During 1921 the monthly earnings and 
expenses of a given railway were as follows: 

Earnings Expenses 


Jan.$19,000* 

Feb. 17,000 

Mar. 22,000 

April. 28,000 

May. 25,000 

June. 24,000 

July. 25,500 

Aug. 25,000 

Sept. 24,000 

Oct. 26,000 

Nov. 20,500 

Dec. 26,000 


*Increase of $1000 over Dec., 1920 


Jan.$20,500f 

Feb. 18,500 

Mar. 19,000 

April. 20,500 

May. 24,000 

June. 23,500 

July . 24,500 

Aug. 23,000 

Sept. 22,500 

Oct. 25,500 

Nov. 20,000 

Dec. 25,000 


fDecrease of $1000 from Dec., 1920 















































224 


PRACTICAL MEASUREMENTS 


2. The decennial public school population (age 5 to 18 yrs.) of 
the United States, and the amount expended for public education, 
since 1878 were as follows: 



Population 

Expenditure 

1878 . 

14,356,000 

79,000,000 

1888 . 

17,827,000 

124,000,000 

1898 . 

21,572,000 

194,000,000 

1908 . 

24,613,000 

371,000,000 

1918. 

27,686,000 

763,000,000 


Show these facts graphical^. 

3 . Of the 27,686,000 persons, 5-18 yrs. of age, in the United 
States in 1918, 20,853,000 were enrolled in school, and the average 
daily attendance was 15,549,000. Show these facts by a bar graph. 

4 . The apple crop of the United States in a recent year was 
147,000,000 bushels. Of that quantity Washington produced 
23,000,000 bushels; New York, 17,000,000 bushels: Virginia, 
10,000,000 bushels; California, 8,500,000 bushels; Pennsylvania, 
8,000,000 bushels; Michigan, 6,500,000 bushels; Missouri, 5,750,- 
000 bushels; Oregon, 5,500,000 bushels. 

Show by means (a) of a circle, and (6) of a bar graph, the rela¬ 
tion of the above values to each other and to the total crop. 

5 . The monthly customs receipts at the New York Custom 
House for the fiscal year ended June 30, 1920, were as follows: (in 
nearest quarter million dollars.) 


1919 


1920 


July. 

$15,250,000 

January . . . 

. . . $21,250,000 

August. 

15,500,000 

February . . . 

. . . 19,250,000 

September. 

16,750,000 

March .... 

. . . 22,250,000 

October. 

16,750,000 

April. 

. . . 20,000,000 

November. 

21,000,000 

May. 

. . . 17,750,000 

December. 

19,250,000 

June. 

. . . 21,250,000 


Prepare a graph of these facts. 



























GRAPHS 


225 


6. Show by a diagram the relative earnings and expenses of a rail¬ 
way whose quarterly earnings and expenses for 1920 were as fol¬ 
lows: 

Earnings Expenses 

Jan.-Mar.$ 3,200* Jan.-Mar.$3,000f 

Apr.-June. 8,200 Apr.-June. 8,900 

July-Sept. 15,500 July-Sept. 8,200 

Oct.-Dec. 5,400 Oct.-Dec. 5,800 

7. Make a diagram showing the production of beet sugar in the 
United States for five consecutive years (in nearest 1000 tons). 

Tons 

1915- 16 874,000 

1916- 17 . 825,000 

1917- 18 . 765,000 

1918- 19 . 761,000 

1919- 20 . 764,000 

8. Illustrate graphically the production of cotton in the United 
States since 1865, as shown by the following (numbers of bales 
given to the nearest 100,000): 

Bales (500 Lb. Each) 

1865 . 2,100,000 

1875 4,300,000 

1885 . 6,400,000 

1895 7,100,000 

1905 . 10,800,000 

1914 . 16,100,000 

1919 . 11,000,000 

9. The total corn crop of the United States for 1919 was 2,917,- 
000,000 bushels. Of this amount, seven states produced as 
follows: 

Iowa. 416,000,000 

Illinois. 301,000,000 

Nebraska. 184,186,000 

Indiana. 175,750,000 

Ohio. 162,800,000 

Missouri. 155,412,000 

Minnesota. 118,000,000 

Show these results graphically. 

♦Decrease from *5200, Oct.-Dec., 1919. fDeoreaBe from *4400, Oct.-Dec. ,1919 

VAN TUYL’S NEW COMP. AR—15 




























226 


PRACTICAL MEASUREMENTS 


The total value of im¬ 
ports of vegetables into the 
United States for a recent 
year was $33,100,000. The 
proportion of the several 
kinds of vegetables is shown 
in the graph at the right. 
Tabulate, as nearly as pos¬ 
sible,. the value of each 
kind. Make estimates in 
round hundred thousands of 
dollars. 



The following graph shows the amount, in long tons, of the 
copper mined in the United States from 1909 to 1919, inclusive. 
Make a tabulation of the number of tons for each year. Esti¬ 
mate values in nearest 10,000 tons. 



From the graph at 
the right tabulate the 
list of cities given, with 
their population. 
Make estimate in near¬ 
est 100,000. 






























































SPEED TEST 


227 


EXAMINATIONS 

SPEED TEST 

Minimum time, thirty minutes; maximum, one hour. Deduct 
one credit for every minute required beyond minimum time. 

514. l. Perform the following operations: 

$1.25 X 28 X 77 X 35 _ 

4 X 11 X 7 X 25 

12' X 16" X 2}" = board feet 

36' X 5' 6" X $1.25 a square yard = $ 

20 planks 16' X 9" X 2§" @ $40 per M = $ 

2. Find the total cost: 3 . Find the number of yards 

of carpet £ yd. wide, for rooms, 
7,480 lb. bran @ $22 per ton. 16' by 14' 

15,870 lb. hay @ 16 per ton. 13' 6" by 12' 4" 

12,456 lb. meal @ 24 per ton. 

9,498 lb. straw @ 12 per ton. 

4. How many double rolls of paper are required for rooms, 

16' X 12' X 8' 6", openings 11 ft. wide? 

18' X 14' X 9', openings 15 ft. wide? 

5. How many paving blocks 12" X 5" are required for streets as 
follows: 

1 \ mi. long, 60 ft. wide? 

| mi. long. 56 ft. wide? 

6. Find weight of each: 

Block of marble 5' X 2' X 1J', sp. gr. 2.72. 

Gold brick 8" X 4" X 2", sp. gr. 19.258. 

7. Find approximate contents: 

Bin 30' X 8' X 6' full of wheat. 

Box 6' X 5' X 4' full of apples. 



228 


EXAMINATIONS 


8. How many gallons are there in a tank of water, 

4 m. long, 2 m. wide, 1.5 m. deep? 

(1 liter = 1.0567 qt.) 

9. How many bushels are there in 1280 Kg. of oats? in 3000 
Kg. of wheat? (1 Kg. = 2.2046 lb.) 

10. Find the number of hectoliters of wheat that can be put into 
a bin 6 m. long, 3 m. wide, 2 m. deep. 

WRITTEN TEST 

515. l. Find the total cost of : 

1350 ft. of pine @ $32.50 per M. 

6240 ft. of hemlock @ 24.50 per M. 

3650 cedar posts @ 9.50 per C. 

4260 lb. of coal @ 7.25 per ton. 

2. A railway train runs 240 rd. in f of miute. Find its veloc¬ 
ity in miles per hour. 

3. How much will it cost to construct a highway 9 mi. 64 rd. 3 
yd. long at $8500 a mile? 

4 . A room in a factory is 65.5 ft. long, 30 ft. wide, and 12J ft. 
high. How many cubic feet does the room contain? 

5. Fresh air enters a room through an opening 8 in. X 14 in., 
with a velocity of 5 ft. per second. At this rate, how many cubic 
feet of air would enter the room in one hour? 

6 . A man had a yard 38 ft. long by 27 ft. wide; he reserved two 
grass plots each 8 ft. square, and had the rest paved with stone at 
$.90 a square yard. How much did the paving cost? 

7 . How much will it cost at 32^ per cubic yard to make an ex¬ 
cavation 40 ft. long, 30 ft. wide, and 9 ft. deep? 

8. A merchant imported 18 hectoliters of almonds at a cost of 
44 pesetas a hectoliter, and sold them at 12 ^ a quart. Find his gain. 

(1 peseta = $.193. 1 liter = .908 dry quarts.) 

9. A vat .95 m. deep, 1.8 m. long, and 1.2 m. wide is full of vine¬ 
gar. Find its value at 20^ a gallon. 

10. A park is 80 rd. long and 56 rd. wide. Around it is a brick 
walk 8 ft. wide. The exposed surface of each brick is 4" X 8". 
Find the cost of the brick at $15 per 1000. 


PERCENTAGE 


516. The term 'percentage includes those subjects in arithmetic in 
the computation of which is taken as the basis of comparison. 
All comparisons of values or quantities are expressed in hun- 
dreths. Instead of the term hundredths , the equivalent Latin 
expression per cent is used. 

Per cent means by the hundred. The per cent sign (%) is 
generally used instead of the words per cent. 

Observe these equations: 

1 = t 2 t& = .25 - 25% 

- = ^ = . 12 * = 12 *% 

8 100 

2 = 661 = ,66f = 66§% 

3 100 

5 _ 38A _ 00 8 _ OO 8 0/ 

13 100 * 38r * " 38t7% 

Note that in each case a common fraction has first been reduced 
to hundredths, and that next the fraction is in its decimal form. 

The last step shows the fraction in its “per cent” form. Note 
also that the per cent form of the fraction is identical with its 
decimal form except that the per cent sign is used instead of a 
decimal point. 

Percentage problems are, therefore, only a continuation of 
problems in fractions under a new name. 

517. Nearly all problems in percentage can be reduced to one of 
the three type forms explained on page 118. In the language of 
percentage problems, the type forms would read thus: 

1. Find 75 % of $48. 

2. 62? % of what number equals $90? 

3. $75 is what per cent of $90? 

229 


230 


PERCENTAGE 


When reduced to their simplest arithmetical form, the types 
appear thus: 

1. 75% of $48 = ? 

2. 62i% of ? = $45. 

3. ? % of $90 = $75? 

In each of these examples there are three elements—the multi¬ 
plicand, the multiplier, and the product. The corresponding per¬ 
centage terms are base, rate, and percentage. 

518. The base is the number or quantity represented by 100%, 
and is the basis of comparison in any given problem. 

519. The rate is the number of hundredths or per cent, and 
shows the ratio of the percentage to the base. 

520. The percentage is the product of the base multiplied by the 
rate. It is the number or quantity whose ratio or relation to the 
base is expressed in the rate. 

521. What number is 24% of 300? 

24% of 300 = 72. 

300 is the base because it is the number with which comparison 
is made. 

24% is the rate because it shows the ratio of the percentage to 
the base. 

72 is the percentage because it is the product of the base and the 
rate. It is the number that is compared with the base. 

522. If the percentage is added to the base, the sum is called the 
amount. 

523. If the percentage is subtracted from the base, the result is 
called the difference. 

524. Since any common fraction can be reduced to an equiva¬ 
lent decimal or per cent form, conversely, any per cent form of a 
fraction can be reduced to an equivalent decimal or common frac¬ 
tion form. 

i = TM = .50 = 50%, and conversely, 50% = .50 = T % = §. 

5 62 1 62- 5 

-= “ = .62^ = 62|%, and conversely, 62|% = .62|=—^=- etc. 


PERCENTAGE 


231 


The student must recognize the difference between 62|% and 

•62|%. The first expression equals .62 \ and the 

100 8 - 

second expression equals .0062J = - 

10000 800 


525. When the per cent sign is used, it takes the place of the 
decimal point, or it is equivalent to moving the decimal point two 
places to the right. 

To change a decimal to a rate per cent, move the point two places to 
the right and annex the per cent sign, and, conversely, 

To change a rate per cent to a decimal, remove the per cent sign and 


move the point two places to the left. 


.045 = 

4.5% = 41% 

15% = .15 


1.25 = 

125% 

li% = .Oli = 

.0125, etc. 

526. Reduce each of the following to the form indicated: 

To the per cent form: 



l. .18 

4. .98 

7. .33* 

10. .125 

2. .25 

5. .05 

8. .62* 

11. .025 

3. .56 

6. .01 

9. .56* 

12. .0025 

To both the decimal and the per cent forms: 


13. i 

16. T V 

19. * 

22. | 

14. i 

17. f 

20. xV 

23. f 

15. | 

18. \ 

21- tV 

24. V 2 

To the decimal form: 



25. 15% 

28. 72% 

31. 12*% 

34. J% 

26. 29% 

29. 125% 

32. 66|% 

35. f% 

27. 4% 

30. 1*% 

33. 37.5% 

36. |% 

To both the decimal form and the common fraction form in its 

lowest terms: 




37. 10% 

40. 75% 

43. \% 

46. 580% 

38. 30% 

41. 87*% 

44. i% 

47. 137*% 

39. 45% 

42. 93f% 

45. i%% 

48. 101*% 

527. In the chapter on Aliquot 

Parts many of the aliquot or frac- 


tional parts of one dollar were learned. 

The aliquot or fractional parts of 100% are the same numeri¬ 
cally as the like parts of a dollar. 



232 


PERCENTAGE 


528. In the following table of fractional equivalents, related 
rates are grouped, the more important rates being in heavy-faced 
type. 

Table of Fractional Equivalents 


Half 

Quarters 

Eighths 

Sixteenths 

Thirds 

Sixths 

Twelfths 

4 

Sevenths 

Ninths 

Miscellaneous 

Rates 

cn 

o 

II 

61% = A 

331% = 1 

141% = i 

11% — A 


m% = a 

66|% = f 

28} % = } 

l|% — A 

25% = i 

3H% = A 


42»% = | 

21% = * 

75% = £ 

43!% = A 

16f% = 1 

57}% = } 

31% = * 


56}% = A 

831% = f 

71}% = ! 

4% = A 

121% = 1 

68f% = « 


85}% = } 

5% = A 

371% = f 

81}% = t! 

81% = tV 

m% = i 

6|% = A 

621% = f 

93}% = it 

411% = A 

22|% = | 

7 +% = * 

871% = l 


581% = iff 

44}% = } 

9tT% = TT 



91f% = H 

55f % = f 

10% - A 




77}% = } 





88|% = » 



529. The labor involved in solving percentage problems is mate¬ 
rially diminished by using the common fractional equivalents of 
rate per cents wherever possible. 

530. Problems in percentage are solved in accordance with the 
fundamental principles of multiplication and division of simple 
numbers, page 117. In the language of percentage they are stated 
thus: 

Principles: l. Base multiplied by the rate = percentage. 

2. Percentage divided by rate = base. 

3. Percentage divided by base = rate. 

531. To find the percentage, the base and rate being given. 

l. Find 43% of $360. 

$360 Base 
.43 Rate 

1080 43% of $360 means .43 of $360. $360 

144 Q multiplied by .43 equals $154.80. 


$154.80 Percentage 











PERCENTAGE 


233 


2. Find 16f % of $540. 


$540 
■ 16f 
360 
3240 
540 
$90.00 


16|% = i 

Or, Multiplying $540 by .16| gives 

l of $540 = $90. * 90 - 0r > since 16 »% = i- take 
& of $540, which equals $90. 


3 . Find !% of $1250. 

8)$12.50 \% means £ of 1%. 1% of $1250 is $12.50; 

$1.5625 = $1.56 atd * of $12 ' 50 = SL56 ' 


4 . What number is 334% greater than $480? 


100% = $480 
334% = 160 
1334% = $640 


The base is always equal to 100%. Hence $480 
is equal to 100%. The required number is 33$%, 
or $, of $480 greater than $480. 33$% of $480 is 
$160. By adding $480 and $160, the desired result, 
$640, is obtained. 


Note. It should be observed that the base and the percentage are like 
numbers. 

532. Performing all operations mentally where possible, find: 


1 . 21% of 200. 

2. 35% of 500. 

3. 124% of 640. 

4. 25% of 700. 

5 . 26% of 700. 

6. 66f % of 960. 

7 . 874% of 1200, 

8. 13% of 300. 

9 . 17% of 400. 

10 . 6% of 475. 

11. 24% of 300. 

12. 16f% of 372. 

13. 8|% of 744. 

14. 18f% of 640. 

15 . 64% of 500. 


16 . 78% of 1000. 

17 . 93% of 800. 

18 . 125% of 960. 

19 . 80% of 1200. 

20 . 75% of 1500. 

21. 6f% of 1800. 

22 . 141% of 2149, 

23 . 221% of 1881, 

24 . 314% of 8000, 

25 . 564% of 3200. 

26 . 250% of 440. 

27 . 375% of 600. 

28 . 2|% of 1600. 

29 . 34% of 1260. 

30 . 14% of 888. 


31 . |% of 84. 

32 . |% of 800. 

33 . 14% of 64. 

34 . 100% of 128. 

35 . 150% of 630. 

36 . 300% of 25. 

37 . 3% of 6140. 

38 . 124% of 8176. 

39 . 584% of 7200. 

40 . 64% of 6560. 

41 . 624% of 6560. 

42 . 34% of 7230. 

43. 334% of 7230. 

44. 3334% of 7230. 

45 . i4j% of 8400. 





234 


PERCENTAGE 


Short Method 


533. Find 28% of $7500. 

j of $2800 = 12100. By the commutative law of multiplication 
(page 117) the product of 28 times 75 equals the 
product of 75 times 28. Therefore, 28% of $7500 equals 75% of $2800. 
75% = f, and f of $2800 = $2100. 


534. In like manner find, mentally, the value of each of the 
following: 


1. 16% of $1250. 

2. 18% of $333|. 

3. 26% of $2500. 

4. 44% of $2500. 

5. 56% of $5000. 

6. 96% of $125. 

7. 160% of $3750. 


8. 180% of $750. 

9. 32% of $375. 

10. 36% of $75. 

11. 240% of $2500. 

12. 320% of $625. 

13. 32% of $3125. 

14. 124% of $6250. 


15. 39% of $666f. 

16. 42% of $166.66§, 

17. 132% of $2500. 

18. 12% of $25. 

19. 36% of $87.50. 

20 . 52% of $75. 

21. 48% of $375. 


PROBLEMS 


535. l. In a certain school the attendance in 1922 was 425. 
What was the attendance in 1923 if there was an increase of 20%? 

2. A house and lot cost $5000. If taxes, repairs, and other 
expenses amount to $180 per annum, what rent per month must 
the owner receive in order to clear 6% on his investment? 

3. If a plant and machinery, valued at $75,000, depreciates at 
the rate of 5% of its value each year, find its value at the end of 
the fifth year. 

4. The capital stock of a manufacturing company is $150,000. 
If the gross receipts amount to 32% of the capital stock, and the 
total expenditures amount to 75% of the receipts, find the profit 
for the year. 

5. If a clerk’s salary is $600 the first year of service, and he 
gets a 10% increase each year for 5 yr., what will be his salary 
the sixth year? 

6. Chemical analysis shows that “Pennsylvania” Portland 
cement contains the following ingredients in the proportion here 
stated: 


PERCENTAGE 


235 


Silica. 22% 

Iron and Alumina.10.98% 

Lime .61.50% 

Magnesia.2.47% 

Sulphur Anhydride.1.70% 

Combustible materials.85% 


Find the amount by weight of each of the ingredients in 5 T. of 
cement. 


7 . The following table shows the per cent of the total of each kind 
of animal exported from the United States in a recent year. Find 
the value of each of the several kinds of animals. 


Kind of Animals 

Per Cent of Total 

Value 

Cattle 

53.646 


Hogs 

5.698 


Horses 

23.796 


Mules 

9.907 


Sheep 

3.082 


All other (including fowls) 

3.871 


Total 

100 . 

$12,003,684 


8. Whole wheat contains the following: 


Water 

12 % 

Protein 

13% 

Fat 

2 % 

Carbohydrates 

71% 

Mineral salts 

2 % 


According to the U. S. Department of Agriculture, the average 
person consumes 5.3 bushels of wheat in a year. Find, to the near¬ 
est tenth of a pound, the weight of each of the substances named 
a person eats in a year in consuming the 5.3 bushels of wheat. 

9. One of the large meat packing companies exported in one 
year 148,487,828 lb. of beef. Four years later the beef exports 
had increased 298.9%. Find the quantity exported in the latter 
year. 














236 


PERCENTAGE 


536. To find the rate, the base and percentage being given. 

What is the ratio of 6 to 24? (See page 128.) 

6 is what part of 24? 

6 is what per cent of 24? or 

What per cent of 24 is 6? 

In each of these examples, 24 is the basis of comparison, or it is 
the base. 6 is the number compared with 24; hence it is the per¬ 
centage. 

The ratio of 6 to 24 is expressed as or In each of the 
examples, the desired relation between the numbers is determined 
by dividing 6 by 24. 6 -r- 24 = J = 25%. That is, the per¬ 

centage divided by the base gives the rate (Prin. 14, page 117). 

537. l. A man’s income is $1800 a year. If he saves $630 
a year, what per cent of his income does he save? 

The problem is of the third type form 
? % of $1800 = $630. as shown in the first equation. Divid- 

$630 -f- $1800 = .35 = 35% ing the percentage, $630, by the base, 

$1800, gives the rate, 35%. 

2. $1500 is what per cent greater than $1200? 

$1500 — $1200 = $300, amount which $1500 is greater than 

$ 1200 . 

$300 = ? % of $1200? 

$300 -5- $1200 = T 3 /«& = i = 25%. 

First find how much $1500 is greater than $1200. The problem then 
becomes, $300 is what per cent of $1200? which, as shown in the solution, is 
25%. 

To the Student. Reduce each problem (either mentally or in writing) 
to its proper type form. Its solution will then be entirely clear to you. 

Use the fractional form of division, when possible; reduce the fraction to 
its lowest terms mentally; and place the result equal to its equivalent rate 
per cent. 


538. Solve, mentally when possible, each of the following: 
What per cent of 


1. 12 is 6? 

2. 18 is 12? 

3 . 200 is 150? 

4 . 72 is 56? 


5 . 25 is 30? 

6. 14 is 21? 

7 . 450 is 225? 

8. 4 yd. is 6 ft.? 


9 . 2 bu. is 3 pk.? 

10. $24 is $2.40? 

11. $1.75 is $.75? 

12. $15 is $.90? 


PERCENTAGE 


237 


What per cent greater than 


13. 54 is 63? 


17. $3000 is $4200? 

18. 3 pk. is 1 bu.? 

19. 1 lb. Troy is 1 lb. avoir.? 

20. 1 oz. avoir, is 1 oz. Troy? 


14. 300 is 390? 

15. 560 is 630? 

16. 270 is 300? 


What per cent less than 


21. 20 is 18? 

22. 30 is 20? 

23. 50 is 40? 


25. 1 oz. Troy is 1 silver dollar? 

26. A long ton is a short ton? 

27. \ is |? 

28. J is i? 


24. 1 bu. is 3 pk.? 


PROBLEMS 


539. l. If in a school there are 160 boys and 192 girls, what 
per cent of the enrollment are boys? girls? 

2. A man asked $180 for his horse. If he accepted $160, what 
per cent less than his asking price did he accept? 

3. A building worth $12,500 is insured for $7500. For what 
per cent of its value is it insured? 

4. In a spelling lesson of 60 words a pupil spelled 3 words 
incorrectly. What per cent of the words were spelled correctly? 

5. A man bought a house for $8000. To pay for it he borrowed 
$4000 on bond and mortgage at 6%. The other $4000 he paid 
from his own money. If the house rents for $800 per annum, and 
he pays $200 a year for taxes, repairs, insurance, etc., besides the 
interest on the mortgage, what rate per cent does the investment 
pay him? 

6. The following table shows the number of organized wage 
earners (males) in New York State employed one or more days 
during the first quarter of a given year. Find, to the nearest tenth 
per cent, the percentage for each limit of time. Prove the work. 


Period 


Number Employed 


80 da. or more 


1-29 da. 
30-59 da. 
60-79 da. 


15,497 

63,695 

189,756 

60,029 






238 


PERCENTAGE 


7 . A merchant fails, having liabilities amounting to $28,500, and 
resources, to $19,000. What per cent of his debts can he pay? He 
owes Thos. Higbee $1846.50. How much will Mr. Higbee receive? 

8. The total amounts of exports of agricultural implements 
from the United States for the six years ending 1919 were as follows: 

1914 .... $21,649,523 1917 .... $33,264,649 

1915 .... $13,582,849 1918 .... $32,849,164 

1916 .... $21,229,774 1919 .... $41,195,494 

Show in per cent the annual increase or decrease of exports. 

9. The number of pupils registered in high schools in Sept., 

1920, in New York City, was as follows: Peh Cent of Total. 


Borough of Manhattan 

29058 


Borough of Brooklyn 

27964 


Borough of Bronx 

8829 


Borough of Queens 

8315 


Borough of Richmond 

1526 


Total 




Find the per cent of the total in each borough. 

10. The average wholesale price per 100 lb. of dressed beef in 
different cities in a recent year was as follows: In New York 
$18.03; Washington $18.41; Chicago $16.33; Detroit $16.74; San 
Antonio $15.80; Atlanta $17.03; Des Moines $16.03; Cleveland 
$16.92; Indianapolis $15.48; Austin $15.19; Fort Worth $15.43. 
Find (a) the average for all the cities; (6) the per cent the highest 
price is above the average; (c) the per cent the lowest price is 
below the average. 

540. To find the base, the percentage and the rate being given. 

18 is J of what number? 

18 is 25% of what number? 

18 is .25 X ? 

The equation is in the form of the second type problem. The 
product (percentage) and one of the factors (rate) are given to find 
the other factor (base). 

18 .25 = 72, the base. 

To find the base, divide the 'percentage by the rate. 







PERCENTAGE 


239 


541. 48 is 18 j% of what number? 

18f% X ? = 48. 

48 -r- .I 84 = 256. Or, 

m% = a . 

xV of the number = 48. 

X V of the number = J of 48 = 16. 

xf of the number = 16 X 16 = 256. 

On reducing the problem to its simplest type form we have 18f % X ? = 48. 
Since a product and one factor are given, the other factor (base) is 
found by dividing the product, 48, by the given factor, 18f%. The result 
is 256. Or, 

By the use of the fractional equivalent of 18f %, the solution can be made 
mentally. 18f% = 

If ^ of the number is 48, A of the number is | of 48, or 16, and xf of the 
number, or the number itself, is 16 times 16, or 256. 

To the Student. Wherever possible, use this fractional method of solu¬ 
tion. It is far shorter and quicker than the solution first explained. The 
first method will have to be used in those problems containing rates that do 
not have convenient fractional equivalents. 


542. Find the number of which 


1 . 

42 is 6 %. 

12. 

240 is 75%. 

23. 

848 is 266|%. 

2. 

18 is 2 %. 

13. 

280 is 624 %. 

24. 

833 is 116|% 

3. 

25 is 20%. 

14. 

224 is 874%. 

25. 

48 is 

4. 

43 is 1%. 

15. 

378 is 56J%. 

26. 

321 is 4%. 

5. 

86 is 43%. 

16. 

663 is 13%. 

27. 

424 is |%. 

6. 

72 is 12 |%. 

17. 

374 is 17%. 

28. 

375 is f%. 

7. 

90 is 6 |%. 

18. 

836 is 19%. 

29. 

15 is - 2 V%- 

8. 

92 is 25%. 

19. 

1236 is 6 %. 

30. 

5j is |%. 

P) 

106 is 334%. 

20. 

360 is 112J%. 

31. 

3§ is 3 %. 

10 . 

125 is 16f%. 

21. 

900 is 128f%. 

32. 

74 is \%. 

11. 

130 is 50%. 

22. 

755 is 250%. 

33. 

44 is §%• 




PROBLEMS 




543. 1 . A young woman, by saving 31J% of her earnings, paid 
for a house and lot costing $1875 in 6 yr. How much did she 
earn a year? 

2. A parlor suite sold for $286, which was 30% more than it cost* 
How much did it cost? 


240 


PERCENTAGE 


3. A bookkeeper receives an increase of 10% in his salary each 
year for four years. His salary for the fifth year is $2196.15. 
What was his salary the first year? 

4. A house rents for $900 a year. The expenses are $250 a 
year. If the net income is 6^%, what was the cost of' the house? 

5. A house was insured for 80% of its value at f %. If the pre¬ 
mium paid was $22.50, find the value of the house. 

6. Of a consignment of eggs, 6% were broken. For the remain¬ 
der, $45.12 was received at 32 ^ a dozen How many dozen eggs 
were in the consignment? 

7. A merchant’s sales for October were $28,460.50, for Novem¬ 
ber were $35,827.43, and for December were $46,921.07. If the 
sales for these three months were 35% of the sales for the entire 
year, find the average monthly sales for the remainder of the year. 

8. A bankrupt can pay 72% of his debts. If his resources 
amount to $13,475.28, what are his liabilities? How much will a 
creditor receive whom he owes $2485? 

9. In an orchard 40% of the trees are winter apples; 10% are 
fall apples; 5% are harvest apples; 30% are pear trees; and the re¬ 
maining trees are peach. If there are 30 peach trees, how many 
trees are in the orchard? 

10. A house and lot cost $6400. The house cost 66f% more 
than the lot. Find the cost of each. 

11. A druggist’s sales increased 15% the sdfcond year, 20% the 
third year, and 25% the fourth year. If thk fourth year’s sales 
were $29,023.13, what were the sales for the first year? 

12. A merchant’s sales increased 16f % the second year, 20% the 
third year, 25% the fourth year, and 14 \% the fifth year. If the 
sales for the five years amount to $213,244.25, what were the sales 
the first year? 

13. A salesman’s sales increased 25% in November over Octo¬ 
ber’s sales, and 20% in December over November’s sales. His 
January sales decreased 40% from those of December. If 
January’s sales amounted to $3501, what were they for October? 

14. In 1920 the population of Okmulgee, Okla., was 17,430, an 
increase of 317.4% in ten years. What was the population in 
1910? 


RAPID DRILL EXERCISE 


241 


REVIEW 


544. RAPID 

DRILL 

l. What is 25% of 36? 

9. 

2. What is 36% of 25? 

10 . 

3. What is 12§% of 64? 

11 . 

4. What is 64% of 12J? 

12 . 

5. What is 8J% of 72? 

13. 

6 . What is 72% of 8J? 

14. 

7. What is 75% of 68? 

15. 

8. What is 68% of 75? 

16. 


17. Of what number is 20, 25%? 

18. Of what number is 25, 20%? 

19. 40 is 10% of what number? 

20. 10 is 40% of what number? 

21. 30 is 15% of what number? 

22. 15 is 30% of what number? 

23. 60 is 75% of what number? 

24. 75 is 60% of what number? 

25. 50 is what per cent of 60? 

26. 60 is what per cent of 50? 

27. 75 is what per cent of 25? 

28. 25 is what per cent of 75? 

29. 20 is what per cent of 50? 

30 . 50 is what per cent of 20? 

31. 20 is what per cent less than 30? 

32. 30 is what per cent more than 20? 

33. 25 is what per cent less than 30? 

34. 30 is what per cent more than 25? 

35. 40 is what per cent more than 25? 

36. 25 is what per cent less than 40? 

VAN TUYL’S NEW COMP. AR.—16 


242 


PERCENTAGE 


545. PERCENTAGE DRILL 



a 

b 

c 

d 

e 

/ 

g 

1. 

$ 15 

$ 18 

$ 10 

25 % 

10 % 

$ 6 

$ 3 

2. 

16 

22 

14 

20 % 

12|% 

3 

4 

3. 

21 

24 

15 

40 % 

30 % 

4.20 

6.30 

4. 

24 

30 

20 

20 % 

33|% 

2 

8 

5. 

32 

40 

28 

16!% 

25 % 

6 

12 

6. 

40 

45 

32 

m% 

1H% 

10 

8 

7. 

48 

56 

36 

33 \% 

25 % 

12 

20 

8. 

50 

60 

45 

25 % 

30 % 

15 

18 

9. 

60 

75 

54 

16|% 

25 % 

20 

18 

10. 

72 

81 

60 

33|% 

16|% 

24 

9 

11. 

80 

96 

70 

20 % 

12|% 

30 

8 

12. 

96 

120 

84 

12|% 

8|% 

24 

16 

13. 

108 

120 

96 

16|% 

5 % 

9 

36 

14. 

120 

144 

100 

37|% 

50 % 

20 

12 

15. 

150 

175 

120 

25 % 

1H% 

50 

40 

16. 

180 

210 

160 

66|% 

m% 

40 

60 

17. 

240 

270 

210 

12*% 

25 % 

60 

30 

18. 

300 

360 

250 

20 % 

25 % 

25 

100 

19. 

360 

420 

320 

33!% 

161% 

90 

45 

20. 

450 

540 

360 

20 % 

16|% 

75 

45 


1. The numbers in column b are what per cent greater than 
those in column a? 

2. The numbers in column c are what per cent less than those in 
column a? 

3. Consider column a as cost, and column d as rate of gain. 
Find the selling price. 

4. Consider column a as cost, and column e as rate of loss. Find 
the selling price. 

5. If column a is cost and column / is gain, find the gain per 
cent. 

6. If column a is cost and column g is loss, find the loss per 
cent. 

7. If column b is cost and column c is selling price, find the rate 
of loss. 














PERCENTAGE REVIEW 


243 


8. If column c is cost and column b is selling price, find the rate 
of gain. 

9. If column b is selling price and column d is rate of gain, find 
the cost. 

10 . If column b is selling price and column e is rate of loss, find 
the cost. 

PERCENTAGE REVIEW 
Mental 

546. 1. Two railroads carry freight 200 mi. The first carries 
it 75 mi., and the second 125 m.. What per cent of the distance 
does each carry it? 

2. A man earns $1500 a year and spends $900. What per cent 
of his money does he spend? 

3. A boy bought a watch for $12 and a chain for $4. Later he 
sold them both for the cost of the watch. What per cent of the 
cost was the selling price? 

4. During the month of May a farmer sells 500 qt. of milk; dur¬ 
ing June he sells 600 qt. What is the per cent of increase? 

5. Of a pile of potatoes containing 120 bu., one bushel out of five 
is bad. What per cent are bad? How many bushels are good? 

6. From a half chest of Oolong tea containing 50 lb., 15 lb. were 
sold one day and 20 lb. the next day. What per cent of the tea re¬ 
mained unsold ? What per cent was sold each day? 

7. A bankrupt's resources are $12,000, and his liabilities are 
$20,000. What per cent of his debts can he pay? He owes Mr. 
Brown $600. How much will he receive? 

8. Last year my house and lot was worth $4000. This year it 
is worth $4400. What per cent has it increased in value? If it 
increases in value at the same rate per cent next year, how much 
will the property then be worth? 

9. In an invoice of 4 doz. crates of oranges, 10% are decayed. 
If there is an average of 8 doz. oranges to a crate, how many 
oranges are spoiled? 


244 


PERCENTAGE 


10. If coffee loses 16% of its weight in roasting, how many 
pounds of roasted coffee will there be in 500 lb. of green coffee? 

11. A clerk’s monthly salary was $80. Owing to hard times, his 
salary is reduced 5%. How much does he now receive per month? 

12. A dairyman had 1000 qt. of milk a week in July. During 
August be increased the amount of grain fed to his cows $3 worth 
per week, and obtained 6% more milk. If he sold the milk at 8 ^ 
a quart, did it pay to increase the amount of grain fed? 

13. If coal has been selling at $12.50 a ton, a 10% increase will 
make it sell for how much? 

14. A merchant’s sales increased 33§% the second year over 
the first year. If the first year’s sales were $72,000, what were the 
sales the second year? 

15. I bought 56 yd. of cloth at $2.40 a yard and sold it to gain 
25%. How much did I receive for all of it? 

16. If merchandise cost $360, and increases in value 16§%, how 
much will it be worth? If later it decreases 16J% in value, how 
much will it then be worth? 

17. In settling with a bankrupt, I received $2400, which was 
75% of my claim. How much did I lose? 

18. A man receives $50 a month rent for a house. His ex¬ 
penses for a year are $150. What is the value of the house if it 
nets him 9%? 

19. During the second year a grocer’s sales increased $5000, 
which was 20% of the first year’s sales. What were the sales each 
year? 

20. If, in problem 19, the profit was 20% of the cost of the 
goods, what were his profits each year? What would have been 
his profits if they were 20% of the selling price? 

21. How much must a young man save so that at 6% he may 
have an annual income of $1200? 

22 . A merchant’s average profit is 12J% of his sales. How 
much merchandise must he sell to make a profit of $6000? 

23. If an enterprise pays 8%, what is the investment, the in¬ 
come being $1600? 


PERCENTAGE REVIEW 


245 


24. A man sold 37|% of his business for $3600. What was the 
entire business worth? 

25. I sold a house and lot at a again of $200, which was 5% of 
the selling price. How much did the property cost me? 

PROBLEMS 

547. l. If potatoes shrink 5% in weight during the winter, what 
is the shrinkage in weight of 1200 bu.? 

2. A store is worth $18,000. It rents for $2000. Repairs cost 
1% of the value of the store; taxes cost 1|%; insurance \% on a 
policy of $10,000; and other expenses amount to $85. The net 
income is what per cent of the gross rental? of the value of the 
store? 


3. A bankrupt paid me 68% of my claim. If I received 
$1275.46, how much did he owe me? 

4. If the duty on certain merchandise was 30%, and the present 
duty is 20% less than formerly, what is the present rate? 

5. What per cent of the total area of the following cities is land? 
What per cent is water? 


Land Area Water Area 


New Haven, Conn, 
Syracuse, N. Y. . 
Scranton, Pa. . 
Fall River, Mass. 
Albany, N. Y. 


11,460 acres 
10,639 acres 
12,362 acres 
21,862 acres 
6,914 acres 


2,880 acres 
309 acres 
147 acres 
3,793 acres 
283 acres 


6. In a school there are 400 girls, which is 14f% more than the 
number of boys. How many pupils are there in the school? 


7. After selling 560 sheep, a shepherd found he had 12J% of 
his flock left. How many sheep were there in the flock? 

8. What is difference in pounds between f of a ton and f % 
of a ton? 


9. A man had $1460 and spent $365 for a year’s board. What 
per cent of his money did his board cost him? The money he spent 
is what per cent of the amount he had left? 

10. An agent sold three farms for $12,750. The price of the sec¬ 
ond was 50% more than the price of the first, and of the third was 
30% less than the price of both the others. Find the price of each. 







246 


PERCENTAGE 


11. A bushel of wheat weighs 60 lb. 5J bu. will make a barrel 
of flour (196 lb.). What per cent of the wheat is flour? 

12. The population New York, N. Y., in 1910 was 4,766,883, 
and in 1920 it was 5,620,048. Find the per cent of increase for 
the ten-year period. 

13 . On an invoice of merchandise valued at $672, a merchant 
paid freight $12.60, and cartage $9.80. The charges were what per 
cent of the value of the goods? 

14 . Find the sum of \% of $893.68; f% of $723.85; *% of 
$3487.60. 

15 . The number of automobiles in the United States for certain 
years were as follows: 

1914 — 1,711,339 
1916 — 3,512,996 
1918 — 6,146,617 
1920 — 7,904,271 

Show the per cent of increase for each two-year period. 

16 * United States exports to and imports from Japan for 2 


years were: 


Exports 

Imports 


1918 

$323,012,765 

$265,064,697 


1919 

$383,190,719 

$414,048,810 


Show in nearest tenth per cent, the per cent of excess of exports 
over imports, or of imports over exports for each year. 

17 . A manufacturer sells to a wholesaler at a profit of 20%; the 
wholesaler to the retailer at a 25% gain; and the retailer to the 
consumer at a gain of 60%. If the consumer pays $14.40, find the 
cost to the manufacturer. 

18 . A house and lot rents for $60 a month. The taxes, in¬ 
surance, repairs, etc., are estimated at $180 per annum. How 
much can a purchaser afford to pay for the property so as to have 
a 6% investment? 

19 . A creditor receives from a bankrupt firm $735 on a debt 
of $1960. What per cent of their debt does the firm pay? If 
their total resources amount to $9474.60, how much are their 
liabilities? 


PERCENTAGE REVIEW 


247 


20. A house and lot cost $6000; the insurance is $15, taxes are 
$60, and repairs are $45 annually. What rent per month must be 
received to realize 7% on the investment? 

21. What per cent of 5 lb. avoirdupois is 7 oz.? 

22. From a cask containing 45 gal., 2 gal. and 1 qt. are lost by 
leakage. What per cent is lost by leakage? 

23 . An agent charged $160 as his commission for selling real 
estate at 2|%. For how much was the property sold? 

24 . A farmer’s crop of grain this year is 15% less than it was 
last year. What was this year’s crop if in the two years he raised 
1554 bu. of grain? 

25 . A drover sold 330 sheep and had 1320 sheep left. What per 
cent of his flock did he sell? 

26 . A speculator increased his real estate holdings by 44%. He 
then sold 25% of all his property, and found he had 810 A. left. 
How many acres had he At first? 

27 . A man drew 65% of his money from the bank and with 40% 
of it bought a house and lot for $5200. How much money had he 
remaining in the bank? 

28 . From his income a young man saved 20%. Had he saved 
$200 more, the per cent saved would have been 30%. What is his 
income? 

29 . A merchant having 1848 yd. of cloth, sold 33^% of it at 
$1.60 a yard, 25% of the remainder at $1.65 a yard, and then what 
was left at $1.55 a yard. How much did he receive for all? 

30 . Multiply each of the following percentages by its relative 
weight, and find the general average by dividing the sum of the 
products by the sum of the relative weights. 

Percentage s 

76.5 
93.25 
87 

91.65 
71.81 


Relative Weights Products 

6 - 

5 - 

4 - 

12 - 

9 - 

General Average- 


248 


PERCENTAGE 


548. The following table gives the average composition of a few 
common food products. 


Food Materials 
(as Purchased) 

Refuse 

Water 

Pro¬ 

tein 

Fat 

Carbo- 

hy¬ 

drates 

Ash 

Food 

Value 

Per 

Pound 

Average 

Cost 

Per 

Pound 

Animal Foods 

% 

% 

% 

% 

% 

% 

Calories 

Cents 

Beef, ribs . . 

20.8 

43.8 

13.9 

21.2 

— 

.7 

1135 

20 

Veal, hind quar. 

20.7 

56.2 

16.2 

6.6 

— 

.8 

580 

18 ’ 

Mutton, loin 









chops . . . 

16.0 

42.0 

13.5 

28.3 

— 

.7 

1415 

18 

Ham, smoked 

13.6 

34.8 

14.2 

33.4 

— 

4.2 

1635 

18 

Chicken . . . 

25.9 

47.1 

13.7 

12.3 

— 

.7 

765 

20 

Mackerel . . 

44.7 

40.4 

10.2 

4.2 

— 

.7 

370 

12 

Oysters . . . 

— 

88.3 

6.0 

1.3 

3.3 

1.1 

225 

18 

Eggs .... 

11.2 

65.5 

13.1 

9.3 

— 

.9 

635 

16 

Butter . . . 

— 

11.0 

1.0 

85.0 

— 

3.0 

3410 

25 

Whole milk 

— 

87.0 

3.3 

4.0 

5.0 

.7 

310 

3 

Vegetable Foods 









Oatmeal . . . 

. — 

7.7 

16.7 

7.3 

66.2 

2.1 

1800 

4 

White bread . 

— 

35.3 

9.2 

1.3 

53.1 

1.1 

1200 

5 

Brown bread . 

— 

43.6 

5.4 

1.8 

47.1 

2.1 

1040 

5 

Beans, dried . 

— 

12.6 

22.5 

1.8 

50.6 

3.5 

1520 

5 

Potatoes . . . 

20.0 

62.6 

1.8 

.1 

14.7 

.8 

295 

1 

Apples . . . 

25.0 

63.3 

.3 

.3 

10.8 

.3 

190 

1 

Almonds . . . 

45.0 

2.7 

11.5 

30.2 

9.5 

1.1 

1515 

40 


549. Energy is produced by fat and the carbohydrates. 

550. Protein is the material of which the tissues are built. 

551. Using the table given above, find the quantity of refuse 
and nutrients of the different classes in the following (to the 
nearest tenth of a pound): 

1. 15 pounds of oatmeal. 5 . 5 pounds of beef ribs. 

2 . 12 pounds of brown bread. 6 . 6 pounds of chicken. 

3 . 1 bushel of potatoes (60 lb.). 7 . 24 pounds of smoked ham. 

4 . 1 gallon of milk (8.6 lb.). 8. 18 pounds of mackerel. 


















PERCENTAGE REVIEW 


249 


The following dietaries have been determined by experiment: 


Quantities per Man per Day 



Protein 

Fat 

Carbohydrates 

Energy Value 


Ounces 

Ounces 

Ounces 

Calories 

Farmers 

3! 

4* 

16| 

3415 

Football players 

8 

12* 

22* 

6590 

Lawyers, teachers, etc. 

4 

4! 

15! 

3220 


Select two kinds of animal food and two vegetables that would be 
suitable for: 

9. A street car motorman. ll. A laboring man. 

10 . A clerk in an office. 12. An athlete. 

Find the quantity of nutrients (protein, fat, and carbohydrates), 
the energy in calories, and the cost of the following daily dietaries: 

13. Beef ribs, 12 oz.; potatoes, 16 oz.; butter, 2 oz.; white bread, 
24 oz.; oatmeal, 2 oz.; eggs, 4 oz. 

14. Beans, 8 oz.; milk, 16 oz.; mackerel, 16 oz.; oatmeal, 4 oz.; 
potatoes, 16 oz.; brown bread, 16 oz.; mutton chops, 8 oz. 

15. Almonds, 2 oz.; brown bread, 24 oz.; chicken, 16 oz.; pota¬ 
toes, 16 oz.; apples, 16 oz.; butter, 4 oz. 

16. Smoked ham, 6 oz.; white bread, 20 oz.; oatmeal, 4 oz.; 
beans, 4 oz.; eggs, 4 oz.; potatoes, 16 oz.; butter, 2 oz. 

17. Prepare a daily dietary containing nutrients in the ratio (ap¬ 
proximately) of protein 1 part, fat 1 part, and carbohydrates 4 
parts, the total amount of nutrients not to exceed 24 ounces. 

18. Find the amount of energy in calories, and the cost of the 
dietary prepared in problem 17. 

19. Prepare a daily dietary containing nutrients in the ratio (ap¬ 
proximately) of protein 1 part, fat 1 part, and carbohydrates 3^ 
parts, the total amount of nutrients not to be less than 24 ounces 
nor more than 32 ounces. 

20. Find the amount of energy in calories, and the cost of the 
dietary prepared in problem 19. 











250 


PERCENTAGE 


BILLING AND TRADE DISCOUNT 

552. A bill or invoice is a formal statement given by the seller 
of merchandise to the buyer. It contains the buyer’s name and 
address; the seller’s name and address; the date of the sale; a list 
of the items sold with the price of each and the total value of 
the entire bill; and the terms of the sale. 

In some cases, the freight, or other transportation charge, is 
paid by the seller, and in other cases, by the buyer. If the buyer 
is to pay transportation charges, the seller frequently prepays it 
and then adds it to the bill. 

Examine the following bill and make a tabulated list of its 
different parts. 


YPashirig’tori, D. G. , sj $ _19 

OTlsis- - 

ssouj,/,?. Hardware Company 

£00 York Avc..N.W 

T< 2 mi s -3//0 


■&s- 

/V.JTO 

n 



tf.ys 


7^ 

7 ^/ 4 ^ 

/■7 s 

/4 



2.2.S-0 

^ 7 

S'0^ 





>2-/~ 


The terms “3/10 n/30” mean that the buyer, if he pays within 
10 days from May 18, may deduct 3% of the amount of the bill, 
but he may, if he chooses, delay making payment until 30 days 
from the date of the bill, at which time he must pay the full 
amount of the bill. N/30 (net 30 days) means no discount allowed 
if paid at end of 30 days. 




















BILLING AND TRADE DISCOUNT 


251 


State, or write, the meaning of the following terms: 5/10, 
3/30, n/60; 1% cash, net 30 days; 6/10, 3/30, n/90. 

Discounts of this kind are called Cash discounts. 

553 . As soon as a bill of merchandise is delivered to the pur¬ 
chaser’s place of business, a stock clerk inspects the goods to see 
whether the right kind and quantity of goods has been received. A 
check mark is placed on the invoice against each item if it agrees 
with the stock received. 

A second checking is necessary to verify the accuracy of the 
price charged for each item, and total cost of the entire invoice. 
If an error is found, the seller is notified at once and an adjust¬ 
ment is made with the buyer. 

The invoice on page 250 properly checked would appear as 
follows: 



If Mr. Cummings decides to pay the above bill on or before 
May 28, he will deduct 3% from the face of the bill and send his 
check for the difference. With the check he will send the invoice 
to be receipted. 




















252 


PERCENTAGE 


The following is a form of the check Mr. Cummings would 
send. 


No. _ 


Cleveund.O^ 

The Exchange bank 


f --.. . * A'Y- 










<2, 






Upon receiving the check and invoice, some authorized person 
of the Phoenix Hardware Company will mark or stamp the 
invoice paid, with date of payment, and sign the Company name 
“by” or “per” his own name or initials. The receipted invoice 
will then be sent back to Mr. Cummings, who will place it on file. 

The following illustration shows the invoice after it has been 
paid and receipted. 


DJ 


Washington, D. C. r 






ai0U3 ^ Phoenix Hardware Company 

Hirms -^> 


York Ave., N.W. 


& 7i*rt^/Ld/ & /<2so 

#=/2- q.ys 

&/*/ /. 7 s 

3 2-2-Jo 

J o^o 

? K\0 BY C ff£ 

-May 2a , , 9 ^ c /f 
"hoenijt Hardware Co 

Par^gy a 


<?7 





7^ 



/*7- 


" 


63 

so^ 

r 2 -/y 

ZS-- 



i 

S2~ . 



2 -/o 

73 

























BILLING AND TRADE DISCOUNT 


253 


554 . The acceptance of a cash discount is usually of advantage 
to the buyer. To illustrate: On an invoice of $1800, with the 
terms 2/10 n/30, and dated June 10, the purchaser would save 2<fc 
on each dollar of the bill, or 2% of its face value, by paying on 
June 20. If he lets the bill run to July 10, only 20 days longer, 
he must pay the full $1800. That is, he makes 2% on the bill 
in 20 days by paying on June 20. There are 18 times 20 days 
in a year. 18 times 2% equals 36%. Hence, by paying early, 
the buyer is getting in the discount the equivalent of 36% interest 
on his money. The buyer might better borrow money, if need be, 
to enable him to pay promptly and secure the benefit of the dis¬ 
count. 

Determine the rate of interest received by the buyer if he takes 
the discount on a bill in which the terms are 1%, n/30; 3/10, 
n/60; 4% cash, n/60; 5% cash, n/90. 

It is of advantage also to the seller if the buyer takes advantage 
of the discount offered. He receives promptly funds due him 
on sales and is not required to have so much capital invested or 
tied up in outstanding accounts. He is spared the expense of 
collecting overdue accounts. The “bad debt” customer is 
generally the one who lets his bills run as long as possible, and 
finally is unable to pay, thus causing serious loss to the seller. 
Hence, the seller can well afford to offer a generous discount as 
an inducement to the buyer to pay promptly. 

555 . It is suggested in the following exercises that one half of 
the class prepare the first invoice and the other half of the the class 
write the second invoice and so on alternately. As a part of a 
class session, have pupils exchange papers so that those who pre¬ 
pared invoice No. 1 will receive invoice No. 2, and vice versa. Let 
each pupil now check the invoice he has received, and if it is 
correct, O. K. it. He will then deduct the cash discount, and 
write a check in settlement thereof. The invoice, properly 
checked and O.Kd., with discount deducted, should then, with the 
check, be returned to the “seller.” If the check is correct as to 
form and amount, the invoice will be receipted and returned to 
the “buyer.” 


254 


PERCENTAGE 


PROBLEMS 

1. The Eureka Grocery Company, Poughkeepsie, N. Y., sold 
to The Kingston Grocery Company, Kingston, N. Y., terms 
3/10 n/30, 

8 doz. Van Camp’s Evaporated Milk at $2.75 
6 doz. Borden’s Evaporated Milk at $2.80 

4 doz. pkg. Cream of Wheat at $2.85 
6 doz. pkg. Quaker Oats at $1.35 

42 cans Crisco, 1 lb. cans, at $.22 
16 cans Crisco, 3 lb. cans, at $.65 
36 lb. Yuban Coffee at $.38 

2. A. B. Johnson & Company, Reading, Pa., bought from 
Gristow Bros., Pittsburgh, Pa., terms 4/10 n/60, 

8 cases, 16 doz., Maryland Tomatoes No. 2 size, at $1.22 per 
doz. 

6 cases, 12 doz., Maryland Tomatoes, No. 3 size, at $1.66 per 
doz. 

48 cans, 8 oz. size, Columbia River Salmon, at $.18 

5 cases, 10 doz., Wisconsin Peas, No. 2 size, at $1.64 per doz. 

3 tub, 156 lb., Pure Leaf Lard, at $.17 

3. McCarthy & Redmond, Joliet, Ill., sold to Mills & Son, 
Louisville, Ky., on account 60 days, with 5% off if paid in 10 days, 

4 doz. No. 16, One piece Dish Pans, at $11.25 
doz. No. 9, Wash Boilers, at $46 

5 doz. No. 11, 1 qt. cups, at $1.15 

3 doz. No. 74, Colonial Percolators, at $48.50 
2 doz. No. 5, Electric Toasters, at $22.50 
f doz. No. 8, Aluminum Tea Kettles, at $56 

4. A. R. Jameson, Vicksburg, Tenn., bought from S. S. Patter¬ 
son, St. Louis, Mo., terms 5/10, n/90, 

16 M No. 5 Envelopes at $2.75 
14 M No. 6 Envelopes at $2.85 
2 C No. 3 Pads, ruled at $6.00 

6 gross Penholders at $3.25 

| gross Superior Writing Fluid at $3.60 


BILLING AND TRADE DISCOUNT 


255 



The number of yards varies in the different pieces. Hence 
the number of yards in each piece must be indicated. The small 
figures indicate quarter yards. Thus, 48 3 means 48J; 51 2 means 
5 If, that is 51J, etc. The total number of yards in one kind of 
cloth is also written in the invoice, thus, the 8 pieces of taffeta 
contain 414f yards, at $2.75 per yard. 


6. Stone & Son, Chicago, Ill., bought of The American Woolen 
Company, Boston, Mass., terms 3/30, n/90, 


8 pcs. Cheviot 43 1 38 2 41 2 42 3 44 3 47 1 39 38 1 at $4.75 

6 pcs. Broadcloth 38 2 37* 39 2 36 2 38 3 37 2 at $5.80 

7 pcs. Serge 47 2 43* 44 3 45* 421 46 2 47* at $5.25 

4 pcs. Jersey Cloth 42 2 47 1 39 2 46 at $2.25 





















256 


PERCENTAGE 


556 . Manufacturers, wholesalers and jobbers issue catalogues 
containing names, descriptions and prices of articles they have to 
sell. The catalogues are sent to retailers and others to whom they 
wish to sell their goods. 

557 . The price quoted in the catalogue is called the catalogue, 
or list, price. It is generally a higher price than it is expected 
the article will sell for. Oftentimes it is the price at which the 
retailer is expected to sell the article. 

558 . In order that the retailer may buy from the manufacturer 
or wholesaler at a price that will enable him to resell the goods 
at a fair price and make a reasonable profit, the wholesaler allows 
him a so-called trade discount from the catalogue price. To 
illustrate: 

The manufacturer's catalogue price of a furnace is $150. He 
sells the furnace to a retail dealer- at a discount of 25% from the 
catalogue price. Find the net cost to the retailer. 

$150 is catalogue price 
25% of $150 = 37.50, the trade discount 

$112.50 is the net price or net cost. 

Find the net amount of the following bills: 


Gross 

Amount 

Trade 

Discount 

Gross 

Amount 

Trade 

Discount 

Gross 

Amount 

Trade 

Discount 

1. 

$ 8 

12|% 

11. 

$ 1.25 

20 % 

21. 

$436 

25 % 

2. 

10 

20 % 

12. 

1.50 

16|% 

22. 

127.50 

16|% 

3. 

12 

25 % 

13. 

3.60 

25 % 

23. 

342.30 

20 % 

4. 

6 

33i% 

14. 

4.50 

10 % 

24. 

98.75 

30 % 

5. 

9 

50 % 

15. 

48 

m% 

25. 

233.60 

40 % 

6. 

14 

14*% 

16. 

45 

m% 

26. 

207.80 

35 % 

7. 

16 

61% 

17. 

72 

8 !% 

27. 

216.75 

33!% 

8. 

20 

10 % 

18. 

50 

4 % 

28. 

328.40 

27!% 

9. 

24 

16|% 

19. 

60 

3 % 

29. 

892.85 

30 % 

10. 

30 

6!% 

20. 

75 

8 % 

30. 

' 943.36 

33!% 


















BILLING AND TRADE DISCOUNT 


257 


The following illustration shows an invoice with the trade dis¬ 
count deducted. 


Jleu> Orleans, La. T sa ip_ 


Anderson Hardware Company 


46£ IPast Street 



3 

33 . 

<77 





C,. 





^ C ^ C ^' 

/s. 

30 





4s. 

/ ro 




<? 


33~y 




v_? o' rfo 



Zj2- 



<?o 


$357 is called the “gross amount” of the bill. 

$249.90 is called the “net amount” of the bill. 

A trade discount is deducted by the seller from the amount of the 
bill before it is sent to the buyer. 

If the General Hardware Co. pays the bill within 10 days of 
its date, they will deduct an additional discount of 3% of $249.90 
for early payment. 

The discount allowed by a wholesaler or manufacturer varies 
from time to time depending on market conditions. If the price 
of an article becomes less because of lower cost of manufacturing, 
the seller offers an additional discount instead of changing his list 
or catalogue price. 

VAN TUYL’S NEW COMP. AR.—17 



















258 


PERCENTAGE 


A manufacturer has been selling clothing to a retail clothier at 
$40 a suit less 25%. Because of a decrease in the cost of materials 
he is able to sell the suit for less money, and offers an additional 
discount of 12§%. What is the cost of the suit to the retailer? 

$40 = list price 

25% of $40 = 10 = 1st discount 

30 = price less 1st discount. 

12|% of $30= 3.75 = 2nd discount 

$26.25 = net cost. 

It is to be noted that when two or more discounts are offered 
on the same bill, the second discount is deducted from the value 
after the first discount has been deducted. A third discount 
would be taken from the amount remaining after the second 
discount is taken off, and so on. 

Bills from which a trade discount has been deducted are subject 
to a cash, or time, discount, the same as illustrated on page 252. 

Two or more discounts offered on the same bill are called a 
‘‘discount series.” 

Find the net cost in each of the following: 


Gross Amount 

Trade Discount 

Gross Amount 

Trade Discount 

1 . 

$60 

20% and 16f % 

11. 

$1145.50 

22\% and 10% 

2. 

75 

20% and 33|% 

12. 

1265.80 

15% and 10% 

3. 

80 

25% and 10% 

13. 

1376.80 

12\% and 5% 

4. 

120 

33|% and 25% 

14. 

1891.84 

16|% and 12*% 

5. 

150 

20% and 10% 

15. 

1938.64 

20% and 16|% 

6. 

240 

20% and 12|% 

16. 

2463.70 

37*% and 25% 

7. 

540 

50% and 20% 

17. 

3284.65 

50% and 12*% 

8. 

840 

37£% and 20% 

18. 

4136.75 

20% and 10% 

9. 

960 

40% and 16f % 

19. 

2348.57 

25% and 5% 

10. 

1200 

50% and-25% 

20. 

3167.86 

33*% and 20% 















BILLING AND TRADE DISCOUNT 


259 


559 . The discount series varies on different articles, hence one 
seller may offer a great variety of discounts. To enable the billing 
clerk to compute the net cost of an invoice quickly, a table of 
Net Cost Rate Factors is used. Such a table shows the net 
cost per cent for each of a larger number of discount series. 


The net cost rate factor is found as follows: 


Assume the discount series is 25% and 20%. 


100 % 

25% 


100 % 

20 % 


75% X 80% 
or 

4) 100% 

25% 


5) 75% 
15% 


Subtract each rate separ¬ 
ately from 100%. Multi¬ 
ply the remainders to- 

60%, N. C. R. Factor. f. ether ' The t pr ° du , c \ is 
the net cost rate factor. 

Or 

Deduct the first discount, 25%, from 
100%, leaving 75%. The second discount, 
20%, is deducted by taking 20% (|) of 
75% from 75%. f of 75% is 15%, which, 
taken from 75%, leaves 60%, the net cost 
rate factor. 


60% N. C. R. Factor. 

The latter method is excellent when the rates of discount are 
familiar aliquot parts of 100%. 


560 . The following table shows the net cost rate factors for a 
limited number of discount series. 


Table of Net Cost Rate Factors 



10% 

121% 

15% 

16!% 

20% 

25% 

33!% 

2f % 

.87750 

.85313 

.82875 

.81250 

.78000 

.73125 

.65000 

5% 

.85500 

.83125 

.80750 

.79167 

.76000 

.71250 

.63333 

10% 

.81000 

.78750 

.76500 

.75000 

.72000 

.67500 

.60000 

10%, 5% 

.76950 

.74811 

.72765 

.71250 

.68400 

.64125 

.57000 

10%, 10% 

.72900 

.70875 

.68850 

.67500 

.64800 

.60750 

.54000 

16|% 

.75000 

.72917 

.70833 

.69444 

.66667 

.62500 

.55556 

20%, 10% 

.64800 

.63000 

.61200 

.60000 

.57600 

.54000 

.48000 

25%, 10% 

.60750 

.59063 

.57375 

.56250 

.54000 

.50625 

.45000 
















260 


PERCENTAGE 


In the preceding table a discount series is made up of a combina¬ 
tion of the rate or rates at the left and the rate at the top of a 
column. Thus, at the left on the 4th line is 10% and 5%, and at 
the top of the first column is 10%. Hence the discount series is 
10%, 10%, and 5%, and its equivalent net cost rate factor is 
.7695, or 76.95%. 

It is often desirable to know what single per cent discount is 
equivalent to a discount series. In every case the single discount 
equal to a series is the difference between the net cost rate factor 
and 100%. That is, the single discount equal to a series of 10%, 
10%, and 5% is 23.05% (100% - 76.95%). 

Some firms prefer a table of single discounts equal to a series 
in place of a net cost rate factor table. 

561 . Using the form and rates given below complete a table 
showing the single discount equal to the indicated series. 

Table of Single Discounts Equal to Discount Series 



20% 

25% 

30% 

331% 

371% 

40% 


5% 

10% 

121% 

15% 

16f, 10% 
20%, 121% 









Find the net cost of an invoice of $1625 less a trade discount 
of 20%, 15% and 10%. 

The net cost rate facor for 20%, 15% and 10% is .612 = 61.2%. 
.612 X $1625 = $994.50, net cost. 

Multiplying the gross cost by the net cost rate factor gives 
the net cost. 

Find the net cost of an invoice of $1860 less 20%, 5% and 2\%. 

100% - 20% = 80% 

100% - 5% = 95% 

100% - 2 \% = 97i% 














BILLING AND TRADE DISCOUNT 


261 


80% of 95% of 97|% = .741, or 74.1%, net cost rate factor. 
74.1% of SI860 = S1378.26 


or by aliquots, 5 

$1860 

372 

20 

1488 

74.40 

40 

1413.60 

35.34 


SI378.26, net cost. 


If the discount series is not found in the net cost rate factor table, deduct 
each rate in the series separately from 100%. Then multiply the gross amount 
of the bill $1860 by the product of the several remainders as illustrated. 

Or, by using aliquots, deduct from the gross amount 1 (20%); then from the 
remainder deduct -jV (5%), and finally from that remainder deduct (2j%). 
The net cost is the same as by the other processes. 


A motor boat listed at $1125 is sold at a discount of 25%, 20% 
and 12J%. How much is the total discount and the net cost? 

The single discount equal to 25%, 20%, and 12J% is 4 ?\%. 

47J% of $1125 = $534.37, trade discount. 

$1125 - $534.37 = $590.63, net cost. 


or by aliquots, 5 


8 


$1125 

225 


900 

225 


675 

84.37 


$590.63 

$1125 - $590.63 = $534.37 trade discount. 


When the amount of the discount is desired, find (from a table 
or by calculation) the single discount equal to the series, and 
multiply the gross amount of the bill by it. The result is the 
trade discount. Deducting the trade discount from the gross 
amount of the bill gives the net cost. 










262 


PERCENTAGE 


Or, by using aliquots, find the net cost, as illustrated, and deduct 
net cost from the gross amount to find the trade discount. 

Note. —In deducting the several discounts of a given series it is immaterial 
which discount is deducted first. It sometimes shortens the calculation to 
change the order of the rates. 

562. The following illustration shows the form of invoice used 
by The J. L. Mott Iron Works. Verify the results given. 


GOODS ALWAYS SHIPPEO AT BUYERS'BISK. NOTWITHSTANDING ANY RELBASE FOR DAMAGE THAT TRANSPORTATION COMPANIES MAY REQUIRE US TO SIGN. 
THESE GOODS WERE SHIPPEO IN PERFECT ORDER, WE HOLD SHIPPING RECEIPT ACCORDINGLY. AND IF THERE BE ANY DAMAGE ON ARRIVAL. 

*. CLAIMS MUST IMMEDIATELY BE MADE AGAINST THE TRANSPORTATION COMPANY. 



PRICES.SUBJECT TO CHANGE WITHOUT NOTICE. 


1 

Thermometer 


$1.13 

1 

Altitude Gauge 


1.81 

15 

Positive Key Air Valves 

.13 

1.95 


1 

3 

4 
7 

3 

4 
7 


12 

Gallon Galv. Expansion’Tank 

^complete 

Net 

12.00 

$4.89 


N.P. Union Elbows 

40/5/21% 

4.13 

12.39 

6.67 

lj" 

Ditto 

3.42 

13.68 


1" 

If 

2.64 

18.48 


i £" 

N.P. Water Had. Valves 

7.38 

22.14 


i*" 

Ditto 

5.29 

21.16 


l" 

in 

3.87 

27.09 




75/10% 

114.94' 

25.87 


37.43 


.12 

$ 37.31 


less freight allowance 441b. at 2 Qf. per cwt. - 












BILLING AND TRADE DISCOUNT 


263 


563 . Make out a bill for each of the following purchases: 
l. A. C. Bolton, Cincinnati, Ohio, bought of the Rochester Hard¬ 
ware Co., Rochester, N. H., Sept. 1, 1924 on account 60 da., 3% 
10 da.: 

24 doz. C. S. Axes @ $10. 

12 doz. Handsaws @ $14. 

Less 20%. 

8 doz. Wrenches @ $12. 

Less 25% and 10%. 



2. B. S. Shenkman & Co., Wheeling, W. Va., bought of Norfolk 
Grocery Co., Norfolk, Va., Dec. 1, 1924, on account 90 da., 2% 

30 da., 5% 10 da.: 

20 mats Java Coffee, 1500, @ 25 j£. 

30 bags Maracaibo Coffee, 3750, @ 20 
30 bags Rio Coffee, 3750, @ 15£. 

A discount of 15% and 10% is allowed on the above. 



















264 


PERCENTAGE 


25 half-chests Japan Tea, 1875 lb., @ 25 

30 half-chests Oolong Tea, 1500 lb., @ 45*f. 

25 half-chests Eng. Breakfast Tea, 15001b., @ 30jf. 

A discount of 20% and 10% is allowed on the tea. Find the net 
amount of the bill if paid on Dec. 11, 1924. 

3. A. J. Cammeyer, New York, bought of New England Shoe 
Co., Boston, Mass., Sept. 18, 1924, on account 30 da., 5% cash: 

100 cases Patent Calf, 3600 pr., @> $2.50. 

80 cases Tan Calf, 2880 pr., @ 2.25. 

60 cases Kangaroo, 2160 pr., @ 2.25. 

120 cases French Kid, 4320 pr., @ 3.50. 

If a discount of 25% and 12|% is allowed on the entire bill, 
find amount required to pay the bill at once. Receipt the bill. 

4. Find the net amount of each of the following: 

160 yd. Silk, @ $3.50. 

120 yd. Broadcloth, @ 2.25. 

240 yd. Duchess Satin @ 2.50. 

80 yd. French Crepon @ 1.75. 

Discount of 33J%, 12^%, and 5%. 

5. 18 grindstones at $2.25; 24 ice cream freezers at $4.50; 4 
doz. jack planes at $7; and 12 doz. mortise locks at $3.50. Dis¬ 
count, 37J%, 25%, and 10%. 

6. 24 boxes lemons at $4.50; 30 boxes oranges at $5.40; 18 
doz. pineapples at $1.25; 20 bunches bananas at $1.25. Discount 
20%, 12i%. 

7. 1800 yd. denim at 12^; 1400 yd. cashmere at 50^; 2000 
yd. kersey at $1.10; 3000 yd. storm serge at 62Discount 
20% and 12*%. 

8. 360 yd. Brussels carpet at $1.20; 800 yd. ingrain carpet at 
75^; 1200 sq. yd. linoleum at 02J0; 900 yd. moquette carpet at 
$1.25. Discount 22J% and 5%. 

9. One dealer offers a piano at $650, less 33*% and 20%. An¬ 
other dealer offers the same grade of piano for $700 less 37J%, 
20%, and 5%. Which is the better offer, and how much? 


BILLING AND TRADE DISCOUNT 


265 


564 . There are many persons who like the following solution 
because of its simplicity. 

1. Find the net cost of goods listed $480, less 25%, 20%, 
and 16f%. 

80 By deducting each rate in the series 

$480 X 3 X 4 X 5 from 100% and reducing the remainders 

-—---— = $240. to their fractional equivalents, and placing 

4X5X0 the resulting fractions as shown in the 

solution, the problem resolves itself into one of simple cancellation. In 
many cases this is a very practical solution. 

2. Which is the better for the purchaser, and how much per cent, 
a series of 30%, 20%, and 10%, or a straight discount of 50%? 
How much on a bill of $325? 

3. A bedroom suite is invoiced at $420 less 20% and 14f%. 
What is the net price? 

4. A gentleman bought an automobile for $2400 less 12§%, 
10%, and 5%. What was the net cost? 

5. Goods listed at $128 are sold subject to a discount of 75%, 
12|%, and 5%. Find the net price. 

6. You are offered 400 suits of ready-made clothing at $18 a suit, 
less 20% and 5%, by A, and at $16 a suit, less 10%, by B. If you 
buy of A, you pay the freight, $4.50. B offers to pay the freight. 
Which offer is better, and how much? 

7. The list price of plows is $12. If you buy them at that price 
less 33J% and 10%, and sell them at the same list price less 20% 
and 5%, how much do you make on 48 plows? 

8. I can buy a bathroom outfit of one dealer for $150, less 
33 \% and 20%. I can get the same outfit of another dealer for 
$125, less 25% and 12 \%. The terms in each case are net 30 da., 
2% cash. What is the least amount of ready money for which I 
can get the outfit? 

9. A merchant bought goods at 30% and 10% less than the list 
price, and sold them at the same list price less 10%. Find his 
profit on goods listed at $1000. 



266 


PERCENTAGE 


10. An overcoat in a clothier’s windows is marked “Was $27.50. 
Now $21.” What is the rate of discount? 

11. The merchants in a certain town gave a ticket of admission 
to the county fair costing fifty cents to each customer who purchased 
$3.50 worth of merchandise. This was equivalent to what rate 
of discount? 

12. The fare one way between two cities is $48.75. If a round- 
trip ticket costs $85, what is the rate of discount? 

13. Which is the better offer, and how much, on a bill amounting 
to $1480, a discount series of 33J% 25%, and 20%, or a series of 
40%, 20%, and 12*%? 

14. A bill of $1260.50 is subject to discounts as follows: $450 
of the bill to 25% and 12*%; $375.60 to 37§% and 8J%; and 
the remainder of the bill to a discount of 16f%. Find the net 
amount of the bill. 

15. Find the net amount of a bill of the following amounts: $80 
less 25% and 20%; $126.50 less 16f% and 12*%; and $172.75 
less 62^% and 25%. The whole bill is subject to a cash discount 
of 5%. 

16. At what price must goods be marked to net $4.80 after allow¬ 
ing a discount of 20%? 

In each of the following, find the list price: 


Net Price 

Discount 

Net Price 

Discount 

17. 

$3.60 

20%, 10% 

22. 

$3.32 

25%, 20%, 16|% 

18. 

$4.20 

25%, 20% 

23. 

$29.70 

25%, 20%, 10% 

19. 

$4.50 

331%, 10 % 

24. 

$28.26 

20%, 15%, 12*% 

20. 

$3.90 

20%, 10%, 5% 

25. 

$52.69 

15%, 10%, 5% 

21. 

$12.14 

25%, 12|% 

26. 

$31.50 

30%, 20%, 10% 


Using the form of statement on page 263 as a model, prepare a 
monthly statement for each of the following accounts, showing 
the amount due at the end of the month in each case. 


BILLING AND TRADE DISCOUNT 


267 


27 . 

J. D. Christopher & Co., Pittsburgh, Pa. 


19— 






19— 





Oct. 

4 

S 

17 

426 

50 

Oct. 

7 

Check C 

24 

225 


14 

S 

24 

327 

60 


18 

Mdse, returned J 

11 

74 


22 

S 

27 

831 

45 


28 

Check C 

30 

500 


28 . 

Mrs. L. B. Ingalls, 22 Sand St. 


19— 






19— 






Nov. 

5 

S 

13 

15 

22 

Nov. 

9 

Cash 

C 

28 

10 


10 

S 

19 

4 

28 


20 

Cash 

C 

32 

5 


15 

s 

23 

7 

50 


25 

1 lb. Coffee 

re- J 




24 

s 

30 

6 

75 



turned 

J 

43 



29 . 

D. E. Lewis, Buffalo, N. Y. 







19- 





9 

S 

43 

1125 


Dec. 

15 

Check C 

92 

900 

17 

s 

51 

1575 

60 


22 

Allowance for 



23 

s 

60 

931 

75 



defects J 

111 

75 

29 

s 

67 

832 

50 


28 

Check C 

107 

1000 


30 . 

E. B. Newell, Richmond, Va. 







19— 





4 

S 

47 

527 

50 

Jan. 

10 

Check C 

120 

300 

11 

S 

53 

673 

85 


17 

Check C 

135 

400 

19 

s 

60 

1147 

80 


24 

Mdse, returned 



27 

s 

67 

931 

75 



J 

170 

125 







30 

Check C 

150 

1000 






































































PROFIT AND LOSS 


565. If eggs are bought at 25ff a dozen and sold at 30^ a dozen, 
what is the profit? The profit is what part of the cost? of the 
selling price? 

If oranges cost 30^ a dozen and are sold to gain 33J% of the cost, 
how much is gained per dozen? What is the selling price per 
dozen? 

An overcoat was sold for $30. How much did the retail clothier 
pay for it if he made a profit of 20% of the selling price? 20% 
of the cost? 

566. The subject of profit and loss has for its object the reckon¬ 
ing of gains, losses, costs and selling prices which arise in business 
transactions. The treatment of the subject will be in two parts: 

First, from the standpoint of merchandising business. 

Second, from the standpoint of the manufacturing business. 

567. By merchandising business is meant a business in which the 
chief purpose is buying and selling merchandise. In such a busi¬ 
ness prime cost is the first or direct cost of the goods. 

568. Gross cost is the prime cost plus all buying expenses, such 
as inward freight, cartage, storage, commission, etc. 

569. Gross cost, then, includes any and every expenditure 
necessary to render the goods ready for sale. 

570. Any expenditure incurred in selling the goods is a selling 
expense and should be deducted from income. Some such ex¬ 
penses are rent, heat, light, salaries and wages of the selling force, 
advertising, delivery expense, depreciation, etc. 

571. Net sales is the difference between the total amount of 
sales and the value of goods returned. 

572. Gross profit is the difference betwen the net sales and the 
gross cost of the goods sold. 

573. Net profit is the difference between the gross profit and 
the total selling expense. 


268 


PROFIT AND LOSS 


269 


Profits are reckoned as per cent of 

a. Gros cost, or 

b. Net sales. 

574. To find the gain or loss and the selling price. 

An article cost a retailer $15. Find his selling price if he makes 
a gross profit of (a) 25% of the cost: (b) 25% of the selling pride. 

(a) 25% of $15 = $3.75 gain (b) Selling price = 100% 

$15 + $3.75 = $18.75 sell- Gain = 25% 

ing price. Cost = 75% 0 f S.P. 

$15 ^ .75 = $20 S. P. 

(b) Since the profit is 25% of the selling price, the selling 
price must represent unity, or 100%. Selling price, in this case, 
is made up of cost and gain. Hence, cost is the difference between 
selling price and gain. Therefore, the cost, $15, divided by 75%, 
gives the selling price, or $20. 

Find the gain or loss and the selling price in each of the 
following: 



Cost 

Gross Profit or Loss on 


Prime 

Cost 

Buy¬ 

ing 

Exp. 

Gross Profit or Loss on 

Cost 

Sales 

Cost 

Sales 

1 . 

$0.20 

20% gain 

20% gain 

13. 

$2.50 

$0.14 

33!% gain 

12% gain 

2. 

.24 

25% gain 

25% gain 

14. 

3.45 

.24 

12!% loss 

18% gain 

3. 

.30 

40% gain 

40% gain 

15. 

4.50 

.06 

8! % gain 

14% gain 

4. 

.32 

12!% gain 

20% gain 

16. 

4.60 

.20 

18f % gain 

20% gain 

5. 

.36 

25% gain 

20% gain 

17. 

6.00 

.10 

50% gain 

22% gain 

6. 

.40 

12!% loss 

20% gain 

18. 

17.00 

.20 

15% loss 

14% gain 

7. 

.48 

16f % gain 

4% gain 

19. 

33.25 

.35 

22!% gain 

16% gain 

8. 

.60 

33 \% gain 

25% gain 

20. 

42.20 

.30 

18% gain 

15% gain 

9. 

.75 

40% gain 

25% gain 

21. 

91.75 

1.25 

37!% gain 

22!% gain 

10. 

.90 

10% gain 

10% gain 

22. 

103.60 

1.40 

40% gain 

30% gain 

11. 

1.12 

37!% gain 

30% gain 

23. 

192.50 

2.50 

45% gain 

35% gain 

12. 

1.20 

15% gain 

25% gain 

24. 

276.20 

2.80 

60% gain 

40% gain 

























270 


PERCENTAGE 


In many cases the buying expense is expressed as a per cent 
of the value of the goods purchased. This percentage is found 
by dividing the total buying expenses for a given period (usually 
a year or more) by the total prime cost of goods for the same 
period. The gross cost of a given article is then found by adding 
to its prime cost the predetermined percentage of expenses. 

Last year a merchants total purchases (prime cost) of mer¬ 
chandise amounted to $328,475.60. His total buying expenses 
(commissions, inward freight, cartage, storage, etc.) were 
$16,422.50. Dividing $16,422.50 by $328,475.60 gives very nearly 
5%. As a result of last year’s experience he estimates that his 
buying expenses this year will be 5% of the prime cost of merchan¬ 
dise. 

Allowing 5% of prime cost as buying expense, find the gross 
cost of articles bought for, 


1. $12 

5. $ .20 

9. $15.80 

13. $125 

17. $225 

2. 

$16 

6. $ .60 

10. $18.40 

14. $140 

18. $250 

3. 

$20 

7. $1.40 

li. $25.60 

15. $128 

19. $275 

4. 

$24 

8. $2.60 

12. $30 

16. $160 

20. $360 


PROBLEMS 

575. 1 . Last year Mr. Amos bought merchandise invoiced at 
$428,750, and paid buying expenses amount to $12,862.50. Using 
that experience as a guide for the year’s business, find the gross 
cost of an article invoiced at $24. 

2. Allowing 4% of purchase price for buying expense, find the 
gross cost and the correct selling price of an article bought for $20, 
and sold to make a gross profit of 20% of the selling price. 

For a period of three years, Mr. Kane’s purchases and buying 
expenses were as follows: 


Purchases Buying Expenses 

1st year.$28,493.50 $1162.64 

2nd year.$36,592.55 $1804.97 

3rd year.$38,694.73 $1721.43 


Using the average percentage of buying expenses for the three 
years, complete the following table: 




PROFIT AND LOSS 


271 



Article 

Prime Cost 

% 

Buying 

Expense 

Gross Cost 

% Profit 
on Sales 

Selling 

Price 

3 . 

Overcoat 

$18 



33 i% 


4 . 

Derby hat 

4 



25% 


5 . 

Neckties 

9 (doz.) 



20% 

Each 

6 . 

Umbrella 

3 



20% 


7 . 

Pr. shoes 

5.60 



16% 


8 . 

Handkfs. 

4.50 (dz.) 



37% 

Each 


9 . A stock of goods valued at $13,500 was damaged by fire and 
water so as to cause a loss of 30%. What was the loss, and for how 
much did the goods sell? 

10. A piece of real estate costing $2400 increased in value 41f%, 
when it was sold. Find the gain and the selling price. 

11. A vegetable dealer buys potatoes at 60^ a bushel and sells 
them at a gross profit of 66f% of the cost. What is the gain 
per bushel, and the selling price per peck? 

12. A fruiterer bought a carload of berries for $1380. He was 
obliged to sell them at a loss of 12|% of the cost. One retailer 
who purchased $375 worth of the berries on credit failed in business 
and paid his creditors 40^ on the dollar. Find the wholesaler’s 
total loss. 

13. A grocer buys sugar at 5^ a pound. He puts it up in bags of 
3| lb. each. At what price per bag must he sell it to gain 2f-% 
of the cost? 

14 . A grocer buys 6 bags Rio coffee, 125 lb. in a bag, at 12^ a 
pound, and 9 bags Java coffee, 75 lb. in a bag, at 20 0 a pound. 
He makes a mixture of the two kinds, and sells it at 30 ff a pound. 
If his cost of doing business is 27.37% of gross sales, find his 
rate of profit, based on sales. 

15 . An invoice of merchandise cost $74.25 less 331% and 10%. 
Freight and cartage were $7.45. Find the gain and the selling 
price if the goods were sold at a gross profit of 37|% of cost. 

16 . A shoe dealer buys shoes at $3 less 16f% and 10% per pair. 
The expenses of the purchase amount to 4% of prime cost. At 
what price must he sell them to clear 15% of sales if selling 
expense is 25% of sales? 













272 


PERCENTAGE 


In a given year a merchant’s selling expenses amounted to 
$21,475.65, and his gross sales amounted to $95,447.33. The 
selling expenses were what per cent of the gross sales? 

$21,475.65 ^ $95,447.33 = .225 = 22§%. 

The total selling expense is frequently spoken of as the “cost 
of doing business.” (C. D. B.) It is expressed as a rate per cent 
and is found by dividing the total selling expense by the gross 
sales. 

The gross cost of an article is $5.25. If the C. D. B. is 22J%, 
find the selling price if a net profit of 7|% of sales is realized. 

The selling price = 100% of itself 

Deductions: 

C. D. B. = 22|% of selling price 
Net profit = 7§% of selling price 

Total deductions = 30% of selling price 

Leaving cost ($5.25) = 70% of selling price 

$5.25 -T- .70 = $7.50, selling price. 

576 . 1 . A dry goods merchant’s gross sales for a given year were 
$225,763.85. His annual statement showed the following items 
charged as selling expenses: Salaries and wages, $31,425; adver¬ 
tising, $2400; delivery expense, $4500; rent, $7200; insurance, 
$500; depreciation, $3500; general expense, $11,000; bad debts, 
$1560. Find his cost of doing business. 

Using the per cent of C. D. B. in the above problem, find the 
selling price of each article in the following list: 



Article 

Prime 

Cost 

% of Buying 
Expense 

Gross 

Cost 

C. D. B. 

% Net 
Profit on 
Sales 

2. 

Silk, yard 

$3.60 

5% 



91% 

3. 

Satin, yard 

2.80 

5% 



12% 

4. 

Shirt Waist 

6.40 

4% 



10% 

5. 

Gloves 

1.75 

8% 



12*% 

6. 

Sweater 

4.75 

6% • 



9% 















PROFIT AND LOSS 


273 


Find the selling price of an article whose gross cost is $3.40, 
to make a net profit of 12% of the cost, if the C. D. B. is 20% of 
the sales. 

The profit is 12% of $3.40 = $.408. 

The cost plus the profit.is $3.40 + .408 = $3,808. 

The selling price is made up of the gross cost + the profit + the 
selling expenses. (C. D. B.) 

That is, the selling price = 100% of itself 
The C. D. B. = 20% of the Selling Price 

The difference (cost + profit) = 80% of the Selling Price 
Hence $3,808 = 80% of Selling Price. 

$3,808 -f- .80 = $4.76, the Selling Price. 


577. Find the selling price of each article: 



Article 

Prime 

Cost 

Buying 

Exp. 

Rate of 
Profit 
on Cost 

Cost of 
Doing 
Business 

1 . 

Fur neckpiece 

$45.00 

8% 

m % 

16!% 

2. 

Sealskin coat 

500.00 

8% 

10% 

12% 

3. 

Auto Robe 

28.00 

7% 

81% 

22% 

4. 

Auto tire 

27.50 

6% 

20% 

20% 

5. 

Handbag 

14.80 

5% 

9% 

15.3% 


To find the gain or loss per cent, the cost and the gain or loss 
being given. 

Find the gain or loss per cent of cost in each of the following: 


Cost 


Cost 


Cost 


1 . 

$0.15 

$0.05 gain 

9. 

$ 3.50 

$ .70 loss 

17. 

$603.00 

$201.00 gain 

2. 

.20 

.05 gain 

10. 

5.00 

1.50 gain 

18. 

74.50 

14.90 loss 

3. 

.36 

.06 loss 

11. 

3.60 

.90 gain 

19. 

38.50 

3.85 gain 

4. 

.45 

.05 gain 

12. 

4.50 

.90 loss 

20. 

94.50 

31.50 gain 

5. 

.50 

.05 loss 

13. 

16.00 

8.00 gain 

21. 

18.00 

22.50 S. price 

6. 

.80 

.20 gain 

14. 

28.00 

4.00 gain 

22. 

27.00 

33.75 S. price 

7. 

1.00 

.20 loss 

15. 

450.00 

75.00 gain 

23. 

40.00 

36.00 S. price 

8. 

2.50 

.40 gain 

16. 

425.00 

170.00 gain 

24. 

64.00 

80.00 S. price 


VAN TUYL’S NEW COMP. AR.—18 































274 


PERCENTAGE 


578. The following cost and sales prices are from a “monthly 
statement of sales” in a hardware store. Find the rate of gain 
in each case. In 1-8 use cost as a basis; in 9-16 use selling price 
as a basis. 


Cost Price 

Sales Price 

Cost Price 

Sales Price 

1 . 

$16.38 

$20.25 

9. 

$ 8.65 

$10.00 

2. 

22.50 

29.45 

10. 

41.50 

48.75 

3. 

12.50 

15.00 

11. 

25.00 

31.25 

4. 

35.25 

41.50 

12. 

54.00 

65.50 

5. 

5.65 

7.50 

13. 

2.50 

3.20 

6. 

4.50 

6.00 

14. 

23.50 

26.65 

7. 

15.25 

20.00 

15. 

21.50 

26.00 

8. 

12.50 

15.00 

16. 

15.00 

18.50 


PROBLEMS 

579. 1. An article cost $4.80, less 16|% and 12J%. Buying 
expenses were 6% of prime cost. It was sold for $5.60 less 12y%. 
If the cost of doing business was 14y%, what per cent of the sales 


was the net profit? 

The net selling price is | of $5.60.= $4.90 

The prime cost is f of J of $4.80 . . . = $3.50 

The buying expenses are 6% of $3.50 . = .21 

The gross cost . =3.71 

The gross profit. =1.19 

The C. D. B. is 14f% of $4.90.= .70 

The net profit.= .49 

The per cent of net profit is $.49 -f- $4.90 = .10 = 10%. 


2. A merchant paid $6.40 less 12|% and 10% for a given article. 
He sells it at $12 less 25%. If his buying expenses are 5%, and 
his cost of doing business is 31.4%, what per cent of the gross cost 
was the net profit? 





















PROFIT AND LOSS 


275 


The net selling price is 75% of $12.=$9.00 

The prime cost is § of of $6.40 . = $5.04 

The buying expense is 5% of $5.04 . = .252 

The gross cost.= 5.292 

The gross profit.= 3.708 

The C. D. B. is 31.4% of $9.= 2.826 

The net profit.= $ .882 


The per cent of net profit is $.882 -f- $5,292 = .16f = 16f%. 

3. Find the rate of gross profit on cost when merchandise is 
bought for $6.40 less 25% and is sold at $6.40 net. 

4. An article cost $120 less 20% and 16f%. Buying expenses 

were 6f%. If cost of doing business was 11|%, what per cent of 
sales was made by selling the article at $120? • 

5. A bookseller bought a book for $8, less 33J% and 10%. If 
his cost of doing business is 13|%, what per cent of sales does he 
make by selling the book for $8, less 10%? 

6. Hats cost $54 a dozen less 20% and 16|%. Buying expenses 
are 5%. Cost of doing business is 27.55%. If the hats are sold 
at $5 each, find rate of profit on gross cost. 

7. A statement of the merchandise account at the close of the 
year showed the following: Inventory at the beginning of the 
year, $3500; purchases $22,500; sales $28,000; inventory at the 
close of the year, $6000. Make a statement showing the gain 
or loss on sales. What per cent of the sales was the profit or loss? 



Returns 

Merchandise Sales 



28000 


Cost 





Merchandise, inv. at beginning 
Merchandise purchases 

3500 

22500 




Merchandise, total cost 

Less inventory at closing 

26000 

6000 




Net cost of sales 



20000 


Merchandise gain 



8000 


$8000 -4- $28,000 =.284 = 284%, rate of profit. 

























276 


PERCENTAGE 


8. At the beginning of the year a merchant had on hand mer¬ 
chandise amounting to $5400. During the year he bought $38,000 
worth, and sold $45,000 worth of merchandise. His buying ex¬ 
penses amounted to $750. His selling expenses were $2250. At 
the end of the year he had on hand an inventory of merchandise 
of $7400. Make a statement showing the gross selling profit; 
the net profit. What per cent of the sales was the gross selling 
profit? the net profit? 

9. Make a statement, showing net gain or loss and find the 
gain or loss per cent, based on gross cost, from the following facts: 
Jan. 1, 1923, inventory $7925; Dec. 31, 1923, purchases for 
the year $53,000; inward freight and cartage $796.50; sales for 
the year $54^837.50; selling expenses $2662.50; December 31, 1923, 
inventory $1721.50. 

580. To find the cost, the gain or loss, or selling price and the 
per cent of gain or loss being given. 

If a profit of 10^ a dozen is made on oranges, and the profit is 
| of the cost, what is the cost? 

If a loss of 3^ a dozen on eggs is a loss of 10% of the selling price, 
what is the cost? 

Find the cost in each of the following: 



Gain 

Gain % 
on Cost 


Gain 

Gain 

% ON 

Selling 

Price 


Selling 

Price 

Gain % 

1 . 

$0.06 

25% 

11. 

$ 12.00 

25% 

21. 

$ 3.75 

16f% loss on C. 

2. 

.07 

20 % 

12. 

15.00 

3|% 

22. 

4.50 

20% gain on S. P. 

3. 

.15 

33*% 

13. 

18.00 

2 *% 

23. 

6.40 

25% gain on S. P. 

4. 

.25 

12 *% 

14. 

30.00 

75% 

24. 

18.00 

33*% loss onC. 

5. 

.45 

16f% 

15. 

160.00 

3H% 

25. 

22.50 

25% less on C. 

6. 

.75 

64 % 

16. 

220.00 

5% 

26. 

37.50 

25% gain on S. P. 

7. 

1.25 

14 % 

17. 

28.00 

33*% 

27. 

31.00 

3*% gain onS. P. 

8. 

2.40 

37*% 

18. 

29.00 

14 % 

28. 

39.00 

2 *% loss onC. 

9. 

3.20 

50% 

19. 

38.00 

66 f% 

29. 

59.00 

l*%loss on C. 

10. 

3.60 

56*% 

20. 

54.00 

9% 

30. 

162.00 

1*% gain on S. P. 




















PROFIT AND LOSS 


277 


A merchant’s cost of doing business is 19.4% of his sales, and his 
net profit is 8.6% of his sales. What should be the gross cost of 
an article which he sells for $75? 


The sales.= 100% of themselves 

The C. D. B. . . = 19.4% of sales 
The net profit . . = 8.6% of sales 

A total of. 28% of sales 

Hence the cost.= 72% of sales 


72% of $75 = $54, cost. 

% 

PROBLEMS 

581. A grocer’s sales amounted to $125,750.60 for a given year. 
His net profit was 11J% of his sales, and his cost of doing business 
was 22J% of sales. Find his gross cost of purchases. 

2. If the buying expenses amount to 8% of prime cost, find the 
prime cost of an article whose gross cost was $28.08. 

3. A merchant’s gross profit was $13,650, which amount was 
26% of the sales. Find the amount of sales, and the gross cost of 
goods sold. 

4. In a special sale a merchant marked a quantity of goods down 
so that his loss was 10% of the cost. If his loss was 75j£ per article, 
what was the selling price of each article? 

5. Rather than carry over a stock of straw hats, a dealer sold 
his stock of hats at one half his former price. His loss on each 
hat was then 25 which was 10% of his original selling price. 
Find the cost of each hat. 

6. My sales for 1 year were $72,375.90. The cost of doing 
business was 17|%, and the net profit was 13|% of sales. Find the 
gross cost of goods sold. 

7. A merchant sold a sideboard at a gain of 25% of the cost, 
which was a gain of $18. When the merchant bought the side¬ 
board, he was allowed discounts of 20% and 10% from the list 
price. Find the net cost and the last price. 

8. A stock of goods was damaged 15% by fire and water. If 
the loss was $12,000, what was the value of the goods? 







278 


PERCENTAGE 


9. A speculator bought a lot and sold it so as to gain 20% of the 
cost. He immediately bought a second lot with the selling price 
of the first, which he afterward sold at a profit of 16f % of its cost. 
If his entire gain was $1400, how much did he pay for each lot? 

10. In selling a quantity of oranges and pineapples, a huckster 
gained $12.50, gaining 25% on the cost of the oranges, but losing 
10% on the cost of the pineapples. If the loss on the pineapples 
was 20% of the net gain, find the cost of each. 

11. I bought a house for 10% below its estimated value and sold 
it for 10% more than its estimated value. My profit was $800. 
How much did I pay for the house? What rate of profit did I 
make? 

12. If you buy at $.485 per pound less 25% and 12f%, and sell 
at the same list price less 16|% and 10%, what rate of discount 
can you allow so that you can have the same rate of profit if you 
have to buy at $.485 less 20% and 10%? 


582. PROFIT AND LOSS DRILL CHART 



Cost 

% Gain 

% Loss 

Gain 

Loss 

Selling 

Price 

Selling 

Price 


1 

2 

3 

4 

5 

6 

7 

1 . 

$ 16.00 

.12| 

.10 

$ 4.00 

$ 1.00 

$18 

$12 

2 . 

18.00 

.16f 

•111 

6.00 

2.00 

24 

15 

3. 

22.50 

.331 

.161 

3.75 

11.25 

30 

15 

4. 

28.00 

25 

.121 

4.00 

7.00 

30 

20 

5. 

36.00 

• Hi 

.25 

12.00 

3.00 

42 

30 

6 . 

48.00 

.081 

.161 

4.00 

2.00 

64 

36 

7. 

62.50 

.20 

.10 

25.00 

12.50 

75 

50 

8 . 

96.00 

.121 

.371 

6.00 

24.00 

128 

80 

9. 

120.00 

.371 

.061 

15.00 

12.00 

160 

108 

10 . 

150.00 

.331 

.04 

15.00 

7.50 

210 

141 

11 . 

240.00 

.15 

.061 

15.00 

24.00 

280 

228 

12 . 

360.00 

.05 

.031 

54.00 

18.00 

450 

324 

13. 

500.00 

.24 

.03 

25.00 

10.00 

575 

490 

14. 

750.00 

.18 

.12 

7.50 

45.00 

900 

720 

15. 

900.00 

.30 

.061 

135.00 

72.00 

1080 

837 

16. 

1200.00 

.081 

.021 

180.00 

84.00 

1400 

1056 

17. 

1500.00 

•06f 

.06 

300.00 

45.00 

1550 

1350 

18. 

1800.00 

.22f 

.02 

450.00 

90.00 

2100 

1500 
















PROFIT AND LOSS 


279 ' 


1. Combine columns 1 and 2, and find selling price. 

2. Combine columns 1 and 3, and find selling price. 

3. Combine columns 1 and 4, and find gain per cent. 

4. Combine columns 1 and 5, and find loss per cent. 

5. Combine columns 1 and 6, and find gain per cent. 

6. Combine columns 1 and 7, and find loss per cent. 

7. Combine columns 2 and 4, and find cost and selling price. 

8. Combine columns 3 and 5, and find cost and selling price. 

9. Combine columns 4 and 6, and find gain per cent. 

io. Combine columns 5 and 7, and find loss per cent. 

REVIEW OF PROFIT AND LOSS 
Mental 

583. l. If milk costs 24^ a gallon, what per cent of the cost is 
gained by selling it at 10a quart? 

2. If eggs cost 50 i a dozen, how many can a grocer sell for 
25^, if he gains 20% of the cost? 

3. When potatoes cost wholesale 80^ a bushel, what per cent 
of the selling price does a retailer make who sells them at 15^ a 
basket (8 baskets to the bushel)? 

4. An article that cost $4 sells for $5. Will it pay to reduce 
the selling price 10%, if by so doing there are three times as many 
sales daily? 

5. A dining table cost $40. Find the selling price to gain 20% 
of the cost; 25%; 15%; 10%; 40%. 

6. An article cost $12. Find selling price to gain 20% of the 
selling price; 25% of the selling price; to lose 20% of the selling 
price; 25% of the selling price. 

7. The cost is $48 per dozen. Find selling price of 1 to gain 25% 
of the cost; of 2 to gain 15%; of 3 to gain 20%; of 2 to lose 5%; 
of 6 to lose 10%. 

8. If chairs cost $72 per dozen and 4 of them sell at $9.00 each, 
and the remainder sell at $7.50 each, what per cent of the cost is 
the profit? 


280 


PERCENTAGE 


9. You buy an article for $4.50 and sell it for $6. If the cost 
drops to $4.20, at what price will you sell to make the same rate 
of profit? 

10. A firm has uncollected accounts of $2400. In closing their 
books for the year, they allow 5% for bad debts. How much do 
they allow for bad debts? 

11. An article which cost $2.50 is sold to gain 20% of the cost. 
Find the gain on 50 such articles. 

12. A dealer bought handsaws at $24 a dozen, less 16f%. What . 
per cent of selling price does he make if he sells them at $2.50 
each? 

13. If coal costs $8 a ton wholesale, and sells at 30% above that 
price, what is the gain on 1000 T., if it costs $1 a ton to deliver it? 

14. The raw material in a rocking chair costs $3.50, the labor of 
making costs $8.50. If the profit is 25% of cost, for how much does 
it sell? 

15. An American bought a horse in Canada for $130. He 
brought him to the United States, paying the duty, $30, and 
sold him for $200. What per cent of the selling price did he gain? 

16. The man (in problem 15) returned to Canada and bought 
another horse with the $200 and paid the duty, 25%. At what 
price must he sell this horse to gain 20% on total cost? 

17. If apples cost 90^ a bushel, and 10% of them spoil, at what 
price per bushel must the remainder be sold to gain 20% on the 
entire cost? 

18. A grocer buys 30 doz. eggs at 40^ a dozen. If, by an accident, 
10% of them are broken, what per cent does he gain or lose by sell¬ 
ing the remainder at 50^ a dozen? 

19. I buy a carpet at $3.60 a yard, less 25% What per cent of 
cost do I make if I sell it at $3.60 net? 

20 . A room is 20' X 15', and requires 360 board feet of matched 
lumber to lay the floor. What is the percentage of waste? 

21. 10 lb. of 15^ coffee are mixed with 10 lb. at 20^. At what 
price per pound should it sell to gain 20% of the cost? 


PROFIT AND LOSS 


281 


22. Equal quantities of 30^ and 40jzf tea are mixed. Find 
selling price to gain 30% of the selling price. 

23. A 40^ tea is mixed with a 70^ grade in the ratio of 2 to 1. 
Find the per cent of profit on cost if the mixture is sold at 65 ^. 

PROBLEMS 

584. l. A merchant bought 440 yd. of cloth for $1100, less 20% 
and 10%. At what price per yard must he sell the cloth to gain 
33|% of the cost? 

2. Hats cost $45 a dozen, less 20% and 16|%. At what price 
should they retail to gain 37f% of the selling price? 

3. If a grocer mixes 28 lb. of one grade of coffee costing 22J^f 
a pound with 32 lb. of another grade costing 17J$£ a pound and 
sells the mixture at 35^ a pound, what per cent of the selling price 
is gained? 

4. I bought goods for $435, less 20%, 12|%, 10%. I sold 
them for $560, less 25%, and 16|%. What per cent of cost did 
I make? 

5. If a desk costs $18 less 16f% and 10%, how many desks can 
be bought for $270? 

6. A fruit dealer bought 60 doz. oranges for $12. He sold 40 
doz. at 30^ a dozen. Of those remaining 5 doz. spoiled. At what 
price must the rest be sold to gain 45% on the cost of all? 

7. The prime cost of a quantity of merchandise was $22,836.54. 
The buying expenses were 4% of prime cost. If the cost of doing 
business was 13J% and the net profit was 7J% of sales, find the 
selling price of the entire quantity of goods. 

8. If 75 of a stock of goods is sold for what the entire stock 
cost, what is the rate of gain on cost? If the remainder of the 
goods is sold at cost, what per cent of cost is gained on the whole 
stock? 

9. At what per cent of the list price should goods be marked 
that are bought at discounts of 25% and 20%, if a profit of 33^% 
of cost is to be realized? 

10. An article cost $24, less 20%, 16f%, and 6J%. What is 
the retailer’s price mark if he marks it to gain 40% of the selling 
price? 


282 


PERCENTAGE 


11. At what price shall each of the items in the following bill be 
sold to gain 25% of cost? 

Boston, Mass., Aug. 3, 1924. 

Mr. James Doyle, 

Pittsfield, Mass. 

Bought of WELLS, FLOWER & CO. 

Terms: Net 60 days. 


6 

Roll Top Desks 

$45.00 

270 



8 

Flat Top Desks 

25.00 

200 



12 

China Closets 

37.50 

450 



9 

Sideboards 

62.50 

562 

50 


12 

Dining Tables 

25.00 

300 



12 

Sets Dining Chairs 

20.00 

240 






2022 

50 



Less 25% and 20% 


809 


1213 


Suggestion. Find at what per cent of the list price the goods must be 
sold to make the required profit. 

12. Find the net amount of the following items, and determine a 
selling price mark for each, if sold to gain 33|% of cost. 

2 doz. Smoothing Planes @ $10.50, less 14^%. 

4 doz. Screwdrivers @ $5.40, less 25% 

200 Carriage Bolts, each f" X 2", @ $5.60 
f" X 2i", @ $5.80 
i" X 4A", @ $6.25, 

less 37|% and 20% 

400 Machine Bolts, f" X 6", @ $8.40, 

less 50% and 10%. 

13. I bought 60 bbl. of apples at $2.25 per barrel. Upon sorting 
them, I found there were 35 bbl. first grade and 21 bbl. second 
grade, the rest being decayed. The first grade sold at $3.50, and 
the second grade at $2.15. If it cost 10^ a barrel to sort, what 
per cent of the selling price was the gain or loss? 

14. By marking goods at $4.50 each, a merchant sold on the 
average 4 of the articles per day. Could he afford to reduce the 
price to $4 if by so doing the sales increased to 8 articles a day, 
the cost being $3.25? What per cent would his profits be in¬ 
creased or decreased each day? 














PROFIT AND LOSS 


283 


15. Dining tables selling for $16.50 at the rate of 6 a day are 
reduced to $14, whereupon the sales increase to 10 a day. If the 
tables cost $10.50, does the merchant gain or lose, and what per 
cent, by the cut in the price? 

16. At a sale the price of a chiffonier was reduced from $28 to 
$21. The regular sales were 3 a day. During the sale there were 
12 sales a day. If the cost was $16.80, were the daily profits 
increased or decreased, and what per cent? 

17. At what price should goods costing $216 be marked to make 
a gross profit of 25% of the cost after allowing a discount of 25%, 
20%, and 10%? 

18. If coal can be bought at $4.75 per long ton and the cost of 
handling is 80 i a short ton, what rate of profit on cost is realized 
if the coal is sold at $6.50 per short ton? 

19. At the beginning of a certain year a firm had on hand $16,800 
worth of merchandise. They bought during the year $47,500 
worth, and paid freight on it amounting to $700. They sold 
$62,500 worth of merchandise and paid freight on it to the amount 
of $850. If at the close of the year they had an inventory of 
$19,500, what rate of profit did they make? Prepare a statement 
showing all essential facts of the merchandise account. 

Find the gain or loss per cent on sales in each of the following: 


20 . Mdse, purchases.$13,250.00 

Mdse, sales. 12,687.50 

Mdse, inventory at closing. 4,893.80 

21 . Mdse, inventory at beginning .... $ 3,890.00 

Mdse, purchases. 10,860.00 

Mdse, sales. 9,640.00 

Mdse, inventory at closing. 6,460.00 

22 . Jan. 1, 1924. Inventory.$ 3,200.00 

Purchases during the year . . . 9,600.00 

Sales during the year. 10,500.00 

Returned to creditors. 450.00 

Returned by debtors. 600.00 

Dec. 31, 1924. Inventory. 4,500.00 











284 


PERCENTAGE 


MANUFACTURING COSTS, PROFITS, ETC. 


585 . A manufacturing business is one which takes raw ma¬ 
terial and converts it into manufactured goods. The goods 
manufactured in a given factory are called its finished product. 
The finished product of one factory may be the raw material of 
another. For instance, a cotton mill may spin the cotton into yarn 
or thread, and weave it into cloth, which is the finished product 
of the mill. A clothing manufactory will take the cloth as its 
raw material and make it into clothing. 

586 . There are several elements or parts entering into the cost 
of manufactured goods. In general, they are: 

1. Prime cost, which consists of (a) cost of the raw material 
entering into the finished product, and (b) the cost of labor em¬ 
ployed directly in the process of manufacture. 

2. Factory expense, which consists of rent, or interest on invest¬ 
ment, depreciation, fuel, heat, light, insurance, foremen’s salaries, 
office salaries, etc. 

3. Selling expenses, which consist of salesmen’s salaries, com¬ 
missions, advertising, or any other expense incurred in making 
sales. 


587. l. A shoe manufacturing company has a special order for 
15,000 pairs of shoes. If their records show the following facts as 
to cost, find the average cost per pair of shoes. 

Raw material cost $18,000 

Direct labor cost 19,000 

Factory expense 


Indirect labor cost $3,000 
Rent 3,000 

Power, heat, etc. 12,000 
Depreciation, repairs 2,000 
Foremen’s salaries 12,000 
Office and adm. exp. 6,000 
Gen. exp., ins., etc. 1,500 39,500 

If the company charged $6 per pair for the shoes, their profits 
were what per cent of their sales? 



MANUFACTURING COSTS AND PROFITS 


285 


What per cent of the total sales was the prime cost? was the 
factory expense? 

2. A clothing manufacturer, in filling an order for 25,000 boys’ 
suits, found his costs as follows: 

Raw material $35,452.60; direct labor $42,138.95; indirect labor 
$7043.65; rent $5052.50; power, heat and light $17,960; de¬ 
preciation on plant and machinery $2632.65; repairs $2456.33; 
office salaries $9500; administrative expenses $2892.75; general 
expenses $2860.57. (a) Find the cost of one suit, (b) The cost 

of raw material was what per cent of the whole cost? (c) Direct 
labor was what part per cent of the total cost? 

If the manufacturer charged a price to net him a profit of 12 J% 
of his total cost, find his selling price of one suit, and his gain on 
the whole lot. His net profit was what per cent of his sales? 

3. A cabinet maker pays for the raw material in a certain chair, 
$6.50. The cost of making is $16.90. His overhead expense is 
15% of his sales. Find his selling price if he clears 10% of his 
sales. 

4 . The raw material in 100 pieces of machinery cost $960. 
There were six separate processes in their manufacture. The 
first required 86 hr. at 75^ per hour; the second 132 hr. at 64^ ; 
the third 164 hr. at 70^ ; the fourth 140 hr. at 68^ ; the fifth 216 hr. 
at 68^; and the sixth 96 hr. at 80^ an hour. Factory expense 
was 26%, and selling expense was 15%, of sales. The machines 
were sold at $30.88 each. The net profit was what per cent of the 
sales? 

5. In a given factory are four departments with floor areas as 
follows: Department A 1920 sq. ft.; B 3240 sq. ft.; C 4320 sq. 
ft.; and D 2880 sq. ft. The overhead expense for the whole 
factory for a year is $24,750. If this expense is divided among 
the several departments according to floor space, find the weekly 
overhead for each department. 


286 


PERCENTAGE 


PROELEMS 

588. A prominent clothing firm recently made public the fol¬ 
lowing facts and figures, based on six months' actual experience, 
to show what becomes of a customer's dollar when spenl; for 

Material.$0.2661 

Labor—Wages for making 

and selling.4862 

Rent.0720 

Taxes.0362 

Miscellaneous: 

“Moneyback” . . . $0.0073 
Delivery and Freight .0072 

Postage, carfare, etc. .0060 

Containers, Twine, etc. .0053 
Printing, Stationery .0050 

Fixture Depreciation .0029 

Building Repairs . . .0018 

Insurance.0016 

Bad Debts.0012 

Telephone.0008 

$0.0391 

Advertising.0183 

Profit—for dividends and surplus .0821 

$1.00 

The same firm gives the following prices of woolen cloth for 


overcoats and suits. the wl? re 1920 

Overcoating per yard .$1.20 $4.55 

Serge for spring suits, per yd.$1.27| $4.50 

A typical wool cloth, per yd.$1.07£ $3.75 


The average number of yards required for an overcoat is 2f, 
for a suit is 3f. 

1. Using the above facts, find (to nearest quarter dollar) 
(a) the selling price of an overcoat in 1920. How much of the sell¬ 
ing price was paid to labor? How much of it was profit? (b) 
the cost to the consumer of a spring suit of serge (1) in 1920; 
(2) before the War. (Assume the same ratio of labor and other 
costs to material before the War as given above.) 

2 . Assume the average weekly cost of telephone service was $28. 
Find the average sales per week; the average net profit per week. 


clothing. 


















MARKING GOODS 


287 


3. A spring suit of serge was made from material costing $4.50 
a yard. Linings, buttons, etc., cost $3.50 for the suit. It was 
priced to. sell at $75. During a special sale suits of this grade 
were sold at a discount of 20%. Using the distribution of costs, 
etc., given on the preceding page, determine the amount of gain or 
loss per suit sold during the sale. 

4. A manufacturing concern sold in one year $109,176.83 worth 
of goods. Of that amount $22,400 was spent for raw material; 
$36,750, for direct labor; $5400, for indirect labor; $8400, for rent; 
$10,500, for advertising; $4500, for miscellaneous expenses; and 
$1575, for bad debts; the balance being profit. Using the illus¬ 
tration on the preceding page as a guide, prepare a schedule 
showing what becomes of the customer’s dollar. Carry results 
out to the nearest tenth of a mill. Prepare a graph illustrating 
the several percentages or ratios. 


MARKING GOODS 


589. It is necessary in many lines of business to have the selling 
price of goods marked on them. When merchants desire to keep 
their buying and selling prices from the public, the price marks are 
written in letters or characters, the interpretation of which de¬ 
pends upon a knowledge of the key to the letters or characters. 
If both cost and selling price are marked, a separate key is used 
for each. 

590. The key consists of any word or phrase having ten different 
letters or characters. 

591. If the price mark contains two or more figures alike in suc¬ 
cession, as $4.55, $7.00, or $5.55, a different letter or character, 
called a repeater , is often used. The repeater makes it more 
difficult for strangers to interpret the marks. 


Cost Key 


Selling Price> Key 


ESROHKCALB NOTSELRAHC 

1234567890 1234567890 

Repeaters W and X Repeaters Y and Z 

The words used are black horse and Charleston, spelled back¬ 
ward. The repeaters are any letters not contained in the key 
with which they are used. 


288 


PERCENTAGE 


592. If the cost of an article is $4.25 and the selling price is 

O.SH 

$5.50 the markings would be as follows: the cost mark 

being above and the selling price mark below a horizontal line. 

593. Using the keys just given, write the cost and selling price 
of each of the following: 


Cost 

Rate of 
Gain 

Cost 

Rate of 
Gain 

Cost 

Rate of 
Gain 

1 . 

$12.00 

25% 

7. 

$22.50 

331% 

13. 

$ 1.20 

16!% 

2. 

6.40 

18*% 

8. 

15.00 

20% 

14. 

1.50 

20% 

3. 

4.50 

20% 

9. 

4.40 

25% 

15. 

2.20 

25% 

4. 

3.30 

331% 

10. 

6.60 

16|% 

16. 

17.00 

25% 

5. 

2.50 

20% 

11. 

8.80 

121% 

17. 

14.00 

20% 

6 . 

1.60 

50% 

12. 

1.10 

10% 

18. 

19.00 

121% 


594. Many kinds of goods are bought and sold wholesale by the 
dozen. The retailer sells them by the piece. Consequently, in 
marking such goods the cost mark shows the cost per dozen, and 
the selling price mark the price of one article. 

In marking goods all calculations as far as possible should be 
mental. In dividing the price of a dozen by 12 to find the cost of 
one article, twelfths of a dollar from tV to will constantly re¬ 
quire reducing to cents. Hence all the twelfths should be memor¬ 
ized. See page 17. 

595. Using the words dozen black, repeaters p and q, as a cost 
key, and what prices, repeaters m and n as a selling price key, write 
the cost per dozen and the selling price per article of each of the 
following: 


Cost per 
Dozen 

Rate of 
Gain 

Cost per 
Dozen 

Rate of 
Gain 

Cost per 
Dozen 

Rate of 
Gain 

1 . 

$12.00 

25% 

6. 

$16.00 

25% 

11. 

$32.00 

121% 

2. 

9.60 

33|% 

7. 

18.00 

331% 

12. 

33.60 

33|% 

3. 

8.40 

25% 

8. 

20.00 

25% 

13. 

34.00 

25% 

4. 

14.00 

20% 

9. 

25.00 

40% 

14. 

45.00 

20% 

5. 

15.00 

40% 

10. 

27.00 

331% 

15. 

56.00 

14*% 
































MARKING GOODS 


289 


Using the same keys, write in figures the price here given: 


Do.kq 

E.zb 

17. -- 

Oq.nk 

1 o _ 

Zl.kq 

IQ 

Z.bk 

W.hp 

te 

H.ps 

19* A 

A.ps 

20. 

ac 

Oz.ok 

Oe.qk 

22. 

Be.kq 

23. —-—- 

Ok.pq 
24.-— 

Dc.ok 
25. - 

H.at 

H.ps 

P.sn 

H.sn 

H.wi 


596. To find the marked price, having given the cost with dis¬ 
count, the rate of gain or loss, and the selling discount. 

A jobber buys bicycles at $24, less 12|%. He sells to gain 33f %. 
What is his marking price if he allows 20% discount ? 
i of $24 = $3, discount. 

$24 — $3 = $21, net cost to the jobber. 

i of $21 = $7, gain. 

$21 + $7 = $28, jobber’s net selling price. 

$28 -f- 80% (f) = $35, jobber’s marked price. 

The discount is £ of $24, or $3, which deducted leaves $21 as the net cost 
of the bicycle. 33£% gain is equal to £ of $21, or $7, making the net selling 
price $28. Since the jobber allows a 20% discount, he sells for 80% of his 
marked price. Hence $28 -5- .80 gives $35, the list price (Prin. 14). 

PROBLEMS 

597. l. A sewing machine catalogued at $40 is sold to a jobber 
subject to a discount of 25% and 20%. Find the jobber’s marked 
price if he sells to gain 25% after deducting 25% and 20%. 

2. A manufacturer sells bathtubs at $35, less 40%. At what 
price should the retailer mark them to sell at 25% discount and 
still make 50%? 

3. A merchant pays $10.80, less 25% and 20%, a yard for lace. 
Find his marking price if he sells to gain 25% after allowing a dis¬ 
count of 10%. 

4 . Find the retail selling price of each of the following items if 
they are sold to gain 33J%: 

4J dozen Ties @ $4.50. dozen Ties @ $15.00. 3 dozen Ties 

@ $10.50. 1J dozen Ties @ $18.00. lj dozen Ties @ $15.75. 
Less 20% and 16|%. 

VAN TUYL’S NEW COMP. AR.—19 










290 


PERCENTAGE 


5. Complete the following invoice, and determine the marking 
price of each chiffonier to clear 20%. (Fosdick pays cash.) 


Peter R. Fosdick, 

New Haven, Conn. 


Cleveland, Ohio, Aug. 10, 1924. 


Bought of OHIO FURNITURE COMPANY 
Terms: n /60 3 /10. 


18 

#5287 Oak Chiffoniers © 22.50 . . 





Less 10% and 5% 





Freight prepaid 



13 


6. Complete the following, and find the retail selling price to 
gain 30%. 

Kalamazoo, Mich., Sept. 14, 1924. 

G. A. Hubbard, 

Chicago, Ill. 


Bought of EDWARDS & CHAMBERLAIN 
Terms: Net Cash 


2 doz. Shovels $12.50 

3 “ Wheelbarrows 15.00 

60 ft. 2" Lead Pipe .25 

Less 20% and 10% 


7. If hats cost $48, less 20% and 16|%, per dozen, what should 
be the list price to gain 35%, and allow 16|% and 10%? 

8. When hosiery costs $9, less 20% and 12J% per dozen pairs 
how shall they be marked per pair to gain 20%, if 12J% discount 
is offered? 

9. A rug 12' by 15' is imported, the price being £5. If the duty 
and other expenses amount to $27.60, and the cost of a pound 
sterling is $4.8665, at what price should the rug sell to make a 
profit of 20% of the selling price? 





















MARKING GOODS 


291 


10 . The list price of saws was $54 per dozen, subject to a dis¬ 
count of 12and 5%; the freight was $2.11. At what price 
must each saw be sold to make a profit of 40% of the cost? 

11. A retail grocer has found by experience that his losses based 
on selling price are 2% from bad debts, 3% from perishable goods 
and 3% from other sources; at what per cent above cost must he 
mark all goods so that, after allowing for the above losses, and for 
12% for cost of doing business, he may still make a profit of 15% 
of the selling price? 

12. An automobile manufacturer lists an automobile at $1275, 
subject to a trade discount of 20% and 16f%. At what price 
must the dealer who buys the automobile sell it in order to gain 
25% of his cost? 

13. A retailer finds that the selling and overhead expenses of 
his business approximate 15% of his total sales. If he buys an 
article for $21.50 less 20% and 10%, at what price must he sell it 
to make a net profit of 25% of the sales? 

14. The cost of manufacturing an article is as follows: mater¬ 
ials $4.72, labor $2.35, and overhead $1.03. At what price must 
the article be sold to realize a profit of 20% of the total cost, and 
also to provide 15% of the selling price for selling expenses? 

15. An implement dealer bought a grain drill for $96 less a 
discount of 10% and 5%. At what price, in nearest whole 
dollars, must he mark the drill, so that he may allow a discount of 
20% from his asking price and still make a profit of 20% of cost? 

16. To manufacture a given article costs $17.50, and to sell it 
costs $6.50. Find the selling price to gain 25%, (a) of cost plus 
selling expense; (b) of selling price. 

17. The wholesale price of lamps is $56 per dozen, less 12|% and 
10%. The freight is $1.40 per dozen. At what price per lamp 
should they be sold to make 16|%'of the selling price? 

18. A machine is in 3 parts, A, B, and C. Its cost is $20. A is 
40% of the cost, B is 35%, and C, 25%. The machine sells at a 
profit of 25% of its cost. If the cost of A is increased 10%, and of 
B 20%, find new selling price to maintain the same rate of profit. 


292 


PERCENTAGE 


COMMISSION AND BROKERAGE 

598. A commission merchant is a person who buys or sells mer¬ 
chandise, or transacts other business, for another person. 

599. The person for whom business is transacted by the com¬ 
mission merchant is called the principal. 

600. Comission merchants usually charge a rate per cent on the 
volume of business transacted. Such a charge is called commission 
or brokerage. In some cases the commission is reckoned at a certain 
price per unit of quantity of merchandise, as the bushel, barrel, etc. 

601. An additional charge called guaranty is sometimes made by 
commission merchants for undertaking to be responsible for sales on 
credit or for the quality and quantity of goods bought, etc. 

602. Merchandise shipped by a principal (also called consignor) 
to be sold by a commission merchant is called (by the consignor) a 
shipment. The commission merchant (also called the consignee) 
calls the merchandise a consignment. 

603. The gross proceeds of a sale or collection is the entire 
amount received by the commission merchant. The net proceeds 
is the amount remaining after all expenses have been paid. 

604. The prime cost of a purchase is the first cost, or the amount 
actually paid for the merchandise. The gross cost is the prime cost 
plus all the expenses of the purchase. 

605. An account sales is an itemized statement of the sales of 
merchandise, the charges thereon, and the net proceeds of the sale. 
The statement is made by the commission merchant and sent to his 
principal. (For illustration, see problem 22, page 294.) 

606. An account purchase is an itemized statement of the quan¬ 
tity, quality, and price of merchandise purchased, the expenses in¬ 
curred, and the gross cost of the purchase. It is made by the pur¬ 
chasing agent and sent to his principal. (See problem 26, page 295.) 

607. The principles of percentage apply in commission. 

The gross sales or prime cosi is the base. 

The rate of commission is the rate. 

The commission is the percentage. 

The net proceeds is the difference. 

The gross cost is the amount. 


COMMISSION AND BROKERAGE 


293 


608. To find the commission, the cost or selling price, and the 
rate of commission being given. 

1. Find the commission and the net proceeds of a sale of mer¬ 
chandise amounting to $4500; commission 5%. 

5% of $4500 = $225, commission. 

$4500 — $225 = $4275, net proceeds. 

The commission is 5% of the gross sales, $4500, which equals $225. The 
net proceeds is the difference between the gross sales and the expenses. 

2. Find the commission and the gross cost of a purchase or mer¬ 
chandise amounting to $3400; commission 4%, guaranty 2%, 
cartage $20. 

4% of $3400 = $136, commission. 

2% of $3400 = $68, guaranty. 

$3400 + $136 + $68 + $20 = $3624, gross cost. 

The commission and the guaranty are found by taking 4% and 2%, 
respectively, of $3400. The gross cost is the sum of the prime cost, $3400, 
and all the charges. 


609. Find the commission and the net proceeds, or the gross 
cost, in each of the following: 



Gross 

Sales 

Com. 

Guar. 

Other 

Expenses 


Prime 

Cost 

Com. 

Guar. 

Other 

Expenses 

1 . 

$2400 

5% 


$15 

11. 

$1030 

4% 

3% 

$8.50 

2. 

3000 

3% 

2% 

25 

12. 

1150 

3% 

2% 

9.60 

3. 

1500 

6f% 

1% 

2% 

13. 

1250 

4% 


12.50 

4. 

1800 

64 % 


$20 

14. 

1440 

61 % 


2% 

5. 

2250 

34% 

3% 


15. 

1880 

121% 


50.00 

6. 

3600 

84 % 


30 

16. 

3360 

16|% 

5% 


7. 

4500 

10% 


28 

17. 

4150 

5% 

21% 


8. 

1600 

24 % 

2% 

12 

18. 

9600 

21% 

21% 

100.00 

9. 

2700 

5% 


50 

19. 

12400 

10% 

21% 

350.00 

10. 

3200 

4% 

2% 

60 

20. 

16800 

121% 

5% 

500.00 


21. An agent sold 950 bbl. pork at $17.50 a barrel, on a com¬ 
mission of 2%. The other charges were for storage $25, freight 
$60, and insurance $5. Find the commission and the net proceeds 






















294 


PERCENTAGE 


22 . Find the net proceeds of the following: 


ACCOUNT SALES HAY 


Sold for Account of 


Albany, N.Y., Oct. 7, 1924. 


August Stadlich, Wellsville, N. Y. 
By F. A. Mead & Co. 


1924 

Sept. 

10 

14 

20 

25 tons Hay 

35 tons Hay 

40 tons Hay 

$15.00 

15.50 

14.75 




Sept. 

6 

20 

Charges 

Freight, $100; Cartage, $75; 

Insurance, $5 

Storage 

Commission, 50^ a ton 

Net proceeds 


180 

10 




23. Prepare an account sales from the following data, using your 
own name as agent, and any name you wish as principal: 

On August 15, you receive a consignment of 3000 bu. of corn. 
The same day you pay freight, 2^ a bushel; cartage, a bushel; 
insurance, $7.50. August 20, you sell 1200 bu. at 60^; August 24, 
900 bu. at 62 ; and August 28, 900 bu. at 61 j£. Your commission 
is 2j£ a bushel, and the storage charge is $15. 

24. A consignment of 460 bbl. of flour was received Nov. 1, 
1924. Freight charges were $56; cartage, $14; cooperage, $3.70; 
advertising, $11.50. November 5, 65 bbl. at $6.40 were sold; 
November 7, 185 bbl. at $6.45; November 9, 50 bbl. at $6.35; and 
November 16, the remainder at $6.50. The storage charge was 
$15.80; commission 2%; guaranty, 1%. Prepare an account sales 
from the foregoing data. 

25. If it requires 5J bu. of wheat to make 1 bbl. of flour, and 
the cost of grinding is 15^ a barrel, and the barrels cost 35 each, 
find the gain or loss per cent to the consignor in problem 24, wheat 
costing 80^ a bushel. 














COMMISSION AND BROKERAGE 


295 


26. Find the gross cost of the following: 

ACCOUNT PURCHASE 

Philadelphia, Pa., Aug. 4, 1924. 

Purchase for the account o 

J. L. Cook, Watkins, N.Y. 

By. H. M. Donovan & Co. 




250 





25 M White Pine Lumber 
30 M Georgia Pine Ceiling 
18 M Hemlock Boards 
22 M White Oak 


Charges 

Cartage 

Commission 2\% 


$72 

90 

32 

96 


Amount charged to your account 


27. Prepare an account purchase from the following facts: 

A. B. Carver & Co., Baltimore, Md., bought for the account of 
Howard Cole of Scranton, Pa., 100 gal. oysters at 75 200 doz. 

clams at ; 600 lb. bluefish at 5j£; and 800 lb. white fish at 8jzL 
Charges are 3% commission and drayage $4.50. 


COMMISSION REVIEW 

Mental 

610. 1 . An agent sold $5000 worth of clothing on a commission 
of 5%. What was his commission? 

2. I bought for my principal sugar worth $1250 on a commission 
of 4%. What was the total cost of the sugar? 

3. A lawyer collected 75% of a claim of $1600 and charged 
10% for collecting. What was his commission? 

4. A book agent’s commission was 40%. He sold 120 books at 
$2.50 each. How much did he earn? 

5. An auctioneer received 5% of his sales, which amounted to 
$2400. Find his commission. 

6 . My agent sold for me 100 tubs of butter averaging 50 lb. each 
at 50^ a pound, and charged 2%. How much did I receive for the 
butter? 

















296 


PERCENTAGE 


Find the proceeds in each of the following: 

7. 20 T. hay at $15; commission 8%; other expenses $25. 

8. 12 T. cabbage at $12.50; commission 6%; other expenses $11. 

9. 300 bbl. flour at $6; commission 3%; storage 3^ a barrel; 
freight 20^ a barrel. 

10. 20,000 ft. lumber at $40 per M; commission 2\% ; freight $30. 

11. I bought through an agent 300 yd. of carpet at $1.50, paying 
him 2%. How much did the carpet cost me? 

12. An agent buys 1000 bu. of wheat at $1.10, and charges \ i a 
bushel. What is his commission, and the total cost? 

13. Find the gross cost of 4000 brick at $15 per M; commission 
50^ a thousand; other charges $2.50. 

Find the gross cost of the following purchases: 

14. 1200 bu. corn at 75; commission 3%; freight \\i a bushel. 

15. 700 T. coal at $12.50; commission 2%; freight $1 a ton. 

16. 800 yd. silk $2.50; commission f%; freight $2.50. 

17. 10 half chests Young Hyson, 700 lb., at 25^ ; commission 2%; 
freight $1.50. 

18. 10 bbl. sugar, 3500 lb., at 5 f; commission 4%; freight and 
dray age $1.25 per barrel. 

19. A. broker bought for me 800 lb. of leather and charged $8, 
commission being at 4%. What was the price of the leather, and 
what was the gross cost including freight at 25 ^ a hundred pounds? 

20. My agent charges 2% commission, and 2% guaranty, for 
buying coffee at 20^ a pound. If the commission and guaranty 
amount to $40, how many pounds were bought, and what was the 
gross cost? 

21. Which is better for the agent, a commission of a bushel 
or 2%, when a bushel sells for 80^ ? 

22. I bought a job lot of muslin consisting of 1500 yd. bleached 
and 800 yd. unbleached for $120; commission 5%; and other 
expenses $9. If I sell the bleached muslin at 7 £ a yard and the un¬ 
bleached at 5ff a yard, what is my profit? 


COMMISSION AND BROKERAGE 


297 


23 . I have 75 bbl. of flour which cost me $360. I send them to 
my agent, who sells them at $6.20, and charges 2% commission, 
and $5.70 for other expenses. What per cent profit do I make? 

24 . Find the per cent profit realized by buying potatoes at 80 ^ 
bushel, and shipping them to New York, if I pay 5^ a bushel for 
freight, 3^ a bushel commission, 2^ a bushel for other expenses, and 
the potatoes sell at $1.10. 

REVIEW PROBLEMS 

611. 1 . An agent remitted to me $247.38 after retaining a com¬ 
mission of 5% for collection. What sum did he collect? What 
was the amount of his commission? 

2. The gross cost of a purchase of sweet potatoes was $864.55. 
The expenses were for dray age $12.50, cooperage $3.35, and com¬ 
mission 2J%. If the potatoes cost $.72 a bushel, how many 
bushels were bought? 

3 . An agent charged his principal $106.25 (commission being 
2J%), for buying 5000 bu. of wheat; the freight charges, etc., 
amounted to $43.75. How much a bushel did the wheat cost the 
principal? 

4 . An agent sold 3000 bu. of wheat, and after deducting his 
commission of 2|%, sent his principal the proceeds, $2808. For 
how much a bushel was the wheat sold? 

5 . A commission merchant sold a consignment of cottonseed oil 
for $12,500 on a commission of 4%. Other expenses of the sale 
amounted to $375. What amount was due to the principal? 

6 . My agent purchased for me 180 bbl. of sugar, averaging 
275 lb. each, at 5 \fi a lb. He charged 2% commission and $71.77 
for other expenses. Find the gross cost of the sugar. If it is sold 
at a pound, what rate of gain is realized? 

7. An agent sold for his principal 6500 bu. of potatoes at $1.20 
a bushel, charging 3 f a bushel for selling. The freight charges were 
2j£ a bushel, and other expenses amounted to $85. He purchased 
for his principal an invoice of silks for $2475, charging 5% for 
buying, and remitted the proceeds. What was the amount of the 
remittance? 


298 


PERCENTAGE 


8. At what price shall I mark an article that cost me $18, so 
that I can instruct my agent to allow 25% discount on the marked 
price, pay him 15% commission for selling, allow 5% of sales for 
bad debts, and still make a profit of 33J%? 

9 . To manufacture a certain style of automobile costs $800. 
What must be the catalogue price of this machine, so that the 
manufacturer can give the retailer 25% discount on the list price, 
pay an agent 16f% commission for selling, and still make a profit 
of 25%? 

10 . A broker sold for me 400 bales of cotton (500 lb. in a bale) at 
12.52 a pound on a commission of $5 per hundred bales. He in¬ 
vested the proceeds of the sale in flour at $5 a barrel, charging 
2% for buying. How many barrels were purchased, and what sum 
was left unexpended in the broker’s hands ? 

Suggestion:— Find the gross cost of 1 barrel of flour before dividing. 

11. An agent received $7500 to invest in apples at $2.40 a barrel 
after paying all. expenses of the purchase. Charges were as fol¬ 
lows: commission 3%; guaranty 2%; dray age 5^ a barrel; and 
freight 12^ a barrel. Find the agent’s commission, the number of 
barrels purchased, and the unexpended balance. 

12. An agent sells $8000 worth of hardware on a commission of 
2|%. He pays freight charges, $162.50; drayage $68.50; and 
charges 2J% for guaranteeing payment. What are the net pro¬ 
ceeds of the sale? 

13 . You are a commission merchant and have received, Sept. 1, 
1924, 840 bbl. of flour to be sold for the Twin City Milling Co. 
You pay the freight, $131.50; drayage $210; advertising $27.50. 
September 8, you sell 225 bbl. at $6.75; September 14, 300 bbl. at 
$6.85; and September 21, the balance at $6.80. Storage charge 
is $16.74, insurance \% of gross sales, and your charge for com¬ 
mission is 2|%. Prepare an account sales, showing all facts here 
given, and the net proceeds. 

14 . You buy through a purchasing agent 42 Ideal refrigerators at 
$17.50; 24 dining tables at $26.40; and 18 dressers at $32.75. 
The agent’s commission charge is 5%, and he pays $15 for drayage. 
Prepare an account purchase using the current date and any 
name of agent and place you wish. 


SPEED TEST 


299 


EXAMINATIONS 

SPEED TEST 


(Minimum time, 30 minutes; maximum, 1 hour): Deduct one credit for 
each minute over minimum time. 


612 . l. Find the selling price of each of the following: 


Cost 

(а) $ 0.16 

(б) 6.00 

(c) 56.00 

(d) 4.50 

(e) 828.00 


37£% gained 
20 % lost 
25 % gained 
16f% lost 
i% gained 


Selling Price 


2. Find in each of the following the number of hours and minutes' 
from the time commenced to time finished: 


Time Commenced 


Time Finished Hours 


Minutes 


(a) 9 : 50 a.m. 

(b) 9 : 16 a.m. 

(c) 9 :45 a.m. 

(d) 10 : 15 a.m. 

(e) 9 :14 a.m. 


4 :13 p.m. 
3 :55 p.m. 
12 :16 p.m. 
1 :00 p.m. 
10 :02 p.m. 


3. Divide the numbers in column 1 by the numbers in column 
2, and express your answer as per cent correct to two figures only: 


Column 1 

Column 2 

Per Cent 

(a) 245,326 

981,304 

— 

( b ) 5,253 

30,900 

— 

(c) 12,596 

59,923 

— 

{d) 257,515 

476,878 

— 

(e) 18,487 

55,460 

— 

4. Multiply each of the following percentages by its relative 
weight, and find the general average by dividing the sum of the 

products by the sum of the relative weights: 


Percentages 

Relative Weights 

Products 

(a) 97.51 

8 

— 

(b) 81.33£ 

12 

— 

(c) 65. 

5 

— 

(d) 69.16 

7 

— 


General average 



300 


EXAMINATIONS 


5. Perform the following in¬ 
dicated operations: 

i of 6876.88 = 

$79,487.98 X 6% = 

9569.64 - £ = 

$6734.17 X.5% = 

7. Find the amount to be re¬ 
mitted to the employer in each 
case: 

Amount Rate of Amount 

Collected Commission Remitted 

(a) $3,400 5 % 

0 b ) 91,500 3£% 

(c) 81,600 2£% 

(d) 30,501 £% 


6. Find the gain per cent in 
each of the following: 

Cost Selling Price Gain % 

(а) $4.50 $4.95 - 

(б) 7,500.00 8,750.00 - 

(c) 5,280.00 7,040.00 - 

(d) 9,600.00 10,200.00 - 

8. Find the net cost of each 
item and total net cost of all: 

List Price Discount Net Cost 

(a) $270.00 25% - 

(b) 480.00 25 % and 20% - 

(c) 6.40 12£% and 10% - 

(d) 960.00 33£%and25%- 

Total 


9. Exports from the United States for two years; find the total 
for each year: 


Article 

Agricultural implements . 
Books, maps, etc. 

Oils, Minerals .... 

Wheat. 

Leather & manufactures of 
Meats and meat products 


1918 

$35,076,911 

11,433,064 

289,040,072 

80,802,542 

100,880,843 

593,924,928 


1919 

$41,195,494 

18,239,016 

328,867,886 

356,898,296 

303,176,539 

1,014,165,889 


Excess 


10 . Find the excess of each kind, and the total excess of exports 
for 1919 over those for 1918 in problem 9. 


WRITTEN TEST 

613. 1 . Harry Fenton, a commission merchant, has sold the 
following consignment of goods for Andrew Crowl: 55 bbl. apples 
at $3.50; 60 bbl. potatoes at $2.75; and 80 bbl. cabbage at $1.50. 
Fenton paid expenses as follows: Freight $17.50; cartage $7.50, 
and charged 5% commission. Find the net proceeds of the con¬ 
signment. 

2. In a given five year period factory accidents in a certain 
state were as follows: Number of men killed 864; number of men 
permanently ‘disabled 6580; and number temporarily disabled 
31,800. Find to the nearest tenth per cent the per cent (a) 
killed, (b) permanently disabled, and (c) temporarily disabled. 











PERCENTAGE 


301 


3. A manufacturing concern can make a piece of machinery at 
a cost of $22.50. To sell it costs an additional amount of $8.75 
The selling price is $50. Find the per cent profit based (a) on 
cost, (b) on selling price. 

4. A lumber dealer’s books show the following records for one 
year: Value of lumber on hand at the beginning of the year 
$15,280; stock purchased during the year $105,450; freight paid 
on purchases $1124.50; sales for the year $134,652.50; value of 
stock on hand at close of the year $21,475. Amount spent for 
labor and delivery expense was $3975.50. Find (a) the cost of 
the lumber sold; (b) the profit on sales; (c) the net profit; and (d) 
the per cent of net profit on total sales. 

5. A man bought a house for $9600. He paid $3600 in cash and 
gave a mortgage at 6% for the remainder of the purchase price. 
The house rented for $90 a month. Taxes averaged $220 a year, 
and other incidental expenses were 12J% of the amount received 
for rent. What rate of income did he receive on his actual invest¬ 
ment? 

6 . A. M. Orton bought of the Excelsior Hardware Co., Dec. 15, 
1924, terms 2/10 net 30, 600 pairs barn-door hangers at $2.75, 
less 25% and 10%; 175 doz. strap hinges at $3.25, less 20% and 
12|%; and 15 doz. wrought iron wrenches at $14.50. Find the 
net cost if paid for Dec. 24, 1924. At what price should these 
items be sold retail to gain 25%? 

7. A coal dealer bought a carload of coal containing 40 long 
tons at $3.60 a ton, and paid freight at $2.60 a ton. He sells the 
coal at $10.50 a short ton. His selling expenses average 30% of 
his total sales. His net profit is what per cent of his gross cost? 
(Express the result in the nearest tenth per cent.) 

8. What must be the list price for an article that cost $20 in 
order to realize a profit of 30% of the cost and still allow a dis¬ 
count of 10% on the asking price? What per cent of the cost will 
be gained or lost if the purchaser becomes bankrupt and pays to 
creditors only 60^ on the dollar? 


INTEREST 


615. If a man lends $1000 for 1 yr. and receives at the end of 
the year $1060, why is he paid $60 more than he lent? 

616. The compensation paid for the use of money is called 

interest 

617. That sum of money for the use of which interest is paid is 
called the principal. In the above illustration, $1000 is the 
principal and $60 is the interest. 

618. The amount of interest to be paid is reckoned as a certain 
rate per cent per annum. The most common legal rate is 6% 
That means that for the use of $1 for a year a man pays 6% of 
$1, or $.06. For the use of $1 for 2 yr. a man pays 2 times 6% 
of $1, or $.12. 

619. The rate of interest is the fractional part of $1 (expressed 
as hundredths, or as a per cent) that is paid for the use of $1 for 
1 yr. 

620. The unit of time used as a basis in interest calculations is 
1 yr. Hence, the expressions 4%, 5%, and 6% interest mean, 
respectively, 4%, 5%, and 6% interest for 1 year's use of a given 
principal. 

621. The sum of the principal and the interest is called the 
amount. 

622. The principles of percentage apply to all solutions of 
interest problems. 

623. Since the rate of interest is a rate for 1 yr., a new element 
—time—has to be taken into consideration in interest computa¬ 
tions. For the purpose of solving problems the rate is equal to the 
rate per cent of interest multiplied by the time expressed as years. 
That is, from a percentage standpoint, if the rate of interest is 6%, 

302 


INTEREST FOR DAYS 


303 


and the time is 2 yr. and 6 mo., the rate is 2\ times 6%, or 15%. 
Hence, 

The principal is equivalent to the base. 

The rate of interest multiplied by the time in years is equivalent 
to the rate. 

The interest is equivalent to the 'percentage. 

The amount is equivalent to the amount. 

624. The legal rate of interest in the several states is determined 
by their legislatures. Valid contracts to pay more than the legal 
rate of interest cannot be made, except in certain cases provided 
for by statute. 

625. Charging more than the legal rate is called usury. Most, 
if not all, the states impose a penalty for charging usury. 

626. There are several kinds of interest,viz., ordinary, accurate,* 
and compound.* 

627. Ordinary interest is simple interest reckoned on a 360-day- 
year basis. 

METHODS 


628. There are a number of good methods of reckoning ordinary 
interest. One method is explained in this work, viz., the banker’s 
60-day method—that being the best adapted to all kinds of 
problems. 

For a preliminary discussion of this method, see pages 32-37. 


629. To find the interest for any number of days. 

Find the interest at 6% on $1660 for 45 da. 

$16 60 = int. for 60 da. at 6%. 

4 15 = int. for 15 da. at 6%. 

$12 45 = int. for 45 da. at 6%. 

Interest for 60 da. is found by pointing off 2 places in the principal, which 
gives $16.60. 45 da. is 15 da. less than 60 da.; hence, the interest for 45 da. is 1 

less than the interest for 60 da. | of $16.60 is $4.15, which, taken from $16.60, 
leaves $12.45. 

DRILL EXERCISE 


630. The student should practice until he can find the total 
interest in each of the following groups in four minutes or less. 
*For definitions, see pages 313 and 333, respectively. 



304 


INTEREST 


the total amount of interest at 6% on: 


631. Find 

l. $1400 for 90 da. 
$1600 for 75 da. 
$1560 for 80 da. 
$1300 for 20 da. 
$1200 for 24 da. 

$ 875 for 50 da. 

$ 930 for 65 da. 

$ 870 for 40 da. 
$1100 for 150 da. 
$1370 for 180 da. 
$ 920 for 36 da. 
$1670 for 42 da. 

4. $1172 for 40 da. 
$1260 for 30 da. 
$1440 for 8 da. 
$1240 for 3 da. 

$ 890 for 45 da. 

$ 680 for 48 da. 

$ 840 for 65 da. 

$ 960 for 55 da. 
$1020 for 70 da. 
$1210 for 90 da. 
$1620 for 80 da. 

$ 740 for 3 da. 


2. $2900 for 75 da. 
$3200 for 45 da. 
$2800 for 15 da. 
$2400 for 21 da. 
$3300 for 25 da. 
$4400 for 33 da. 
$1800 for 85 da. 
$1200 for 55 da. 
$1000 for 72 da. 
$2000 for 84 da. 
$3000 for 80 da. 
$4000 for 48 da. 

5. $ 720 for 14 da. 
$ 840 for 16 da. 
$ 960 for 18 da. 
$1200 for 21 da. 
$1500 for 27 da. 
$1800 for 28 da. 
$3600 for 36 da. 
$7200 for 54 da. 
$5400 for 45 da. 
$6300 for 75 da. 
$4500 for 90 da. 
$7500 for 12 da. 


3. $7340 

for 

6 da. 

$8970 

for 

12 

da. 

$5980 

for 

3 da. 

$3890 

for 

9 da. 

$9660 

for 

10 

da 

$4520 

for 

7 da. 

$3360 

for 

5 da. 

$2180 

for 

9 da. 

$1890 

for 

10 

da. 

$3280 

for 

18 

da. 

$4170 

for 

2 da. 

$7320 

for 

1 da. 

6. $ 

80 

for 

15 

da. 

$ 

90 

for 

60 

da. 

$ 

60 

for 

90 

da. 

$ 

72 

for 

45 

da. 

$ 

45 

for 

72 

da. 

$ 

40 

for 

45 

da. 

$ 

300 

for 

20 

da. 

$ 

200 

for 

30 

da. 

$ 

180 

for 

21 

da. 

$ 

210 

for 

18 

da. 

$ 

360 

for 

40 

da. 

$ 

500 

for 

12 

da. 


632. To find the interest for years, months, and days. 

l. Find the interest at 6% on $840.50 for 5 mo. 10 da. 


int. for 4 mo. at 6%. 


Interest for 2 mo. is 
found by pointing off two 
'places in the principal, 4 
mo. is twice 2 mo., and 
the interest for 4 mo. 
is twice $8,405, or $16.81. 
Similarly, the interest for 
1 mo. is one half of $8,405, 

or $4.2025 and the interest for 10 da. is £ of $4.2025, or $1.4008 On adding, 
$22.41 is found to be the interest for the entire time. 


$ 8 

40.50 

16 

81.00 

4 

20 25 

1 

40 08 

$22 

41 33 


41 33 = int. for 5 mo. 10 da. at 6%. 





INTEREST FOR YEARS, MONTHS, AND DAYS 


305 


2. Find the interest at 6% on $1360.75 for 1 yr. 7 mo. 25 da. 


= int. for 20 mo. at 6%. 
1339 = int. for 5 da. at 6%. 


1 yr. 7 mo. 25 da. 
= 19 mo. 25 da. = 
20 mo. less 5 da. 
From the interest for 
2 mo., it is easily 
found for 20 mo. to 
be $136,075. 20 mo. is 5 da. too much; hence, find the interest for 5 da., 
$1.1339, and deduct it from $136,075, leaving $134.94, the desired result. 

3. Find the interest at 6% on $1500 for 2 yr. 3 mo., 23 da. 


$13 

60.75 

136 

075 

1 

1339 

$134 

9411 


9411 = int. for 1 yr. 7 mo. 25 da. at 6%. 


The time is 27 mo. 
23 da. 13 times the inter¬ 
est for 2 mo. gives the 
interest for 26 mo. The 
interest for the remaining 
1 mo. is £ of $15, or $7.50. 
20 da. is f of 60 da.; 
hence, the interest is | of 
$15, or $5. For 3 da. the 
interest is to of the interest for 1 mo., or $.75. The total is $208.25. 

Note. Keep partial results to the fourth decimal place to insure accurate 
results. 


$15 

00 : 

195 

00 : 

7 

50 : 

5 

00 : 


75 

$208 

25 


int. for 2 mo. at 6%. 
int. for 26 mo. at 6%. 
int. for 1 mo. at 6%. 
int. for 20 da. at 6%. 
int. for 3 da. at 6%. 

== int. for 2 yr. 3 mo. 23 da. at 6%. 


633 . Find the interest at 6% on the following: 


Principal 

Time 

Principal 

Time 

1. $ 38.40 

2 mo. 15 da. 

14. $560 

45 da. 

2. $126.75 

4 mo. 20 da. 

15. $640 

29 da. 

3. $248.50 

6 mo. 12 da. 

16. $832 

24 da. 

4. $360 

8 mo. 15 da. 

17. $975 

18 da. 

5. $375.60 

9 mo. 20 da. 

18. $760 

19 da. 

6. $890.50 

11 mo. 10 da. 

19. $480 

1 mo. 18 

7. $429.60 

1 yr. 4 mo. 15 da. 

20. $760 

3 mo. 20 

8. $387.50 

2 yr. 6 mo. 20 da. 

21. $1840 

9 da. 

9. $840 

90 da. 

22. $1920 

28 da. 

io. $660 

100 da. 

23. $1500 

42 da. 

ll. $930 

4 mo. 

24. $7290 

48 da. 

12. $740 

6 mo. 

25. $4000 

54 da. 

13. $425 

30 da. 

26. $3000 

50 da. 


VAN TUYL’S NEW COMP. AR.—20 








306 


INTEREST 


Principal 

Time 

27. $125 

5 mo. 15 da. 

28. $115.75 

9 mo. 13 da. 

29. $825.40 

7 mo. 14 da. 

30. $1260 

3 mo. 23 da. 

31. $1340 

5 mo. 29 da. 

32. $1696 

1 mo. 20 da. 

33. $1440 

70 da. 

34. $5000 

65 da. 

35. $2840 

36 da. 

36. $1000 

75 da. 

37. $2375 

90 da. 

38. $1024 

95 da. 

39. $2717 

7 mo. 15 da. 


Principal 

Time 

40. $582 

1 yr. 3 mo. 5 da 

41. $2135 

2 yr. 6 mo. 

42. $4240 

10 mo. 11 da. 

43. $1325 

9 mo. 8 da. 

44. $2150 

5 mo. 13 da. 

45. $6322 

8 mo. 15 da. 

46. $4178 

132 da. 

47. $7500 

96 da. 

48. $8800 

84 da. 

49. $8100 

78 da. 

50. $5400 

154 da. 

51. $4300 

150 da. 

52. $3300 

165 da. 


634. To find the interest at any rate per cent. 


What is the interest on $4800 for 60 da. at 6%? at 5%? at 
4%? at 7%? 


^—✓ 

m 

00 

00 int. at 6% 

8 

00 int. at 1% 

$40 

00 int. at 5% 


(6) $48 00 int. at 6% 
16 00 int. at 2 % 
$32 00 int. at 4% 


(c) $48 

_8 

$56 


00 int. at 6% 
00 int. at 1% 
00 int. at 7% 


In solution (a), the interest at 6% is found by pointing off two places. Since 
5% is 1% less than 6%, the interest at 5% is found by subtracting the interest 
at 1% from the interest at 6%. To find the interest at 1%, divide the interest 
at 6% by 6. On subtracting, the interest at 5%, $40, is found. 

In solution (6), the interest at 2% is subtracted from the interest at 6% to 
find the interest at 4%. The interest at 2% is found by dividing $48 by 3. 
Why? 

In solution (c), the interest at 1% is added to the interest at 6% to find the 
interest at 7%; and so on. 

Note. The student should observe that in each case the interest is first 
found for the given time at 6%. Then the interest at 6% is increased or dimin¬ 
ished in proportion as the required rate is greater or less than 6%. 








INTEREST AT ANY RATE 


307 


635. The interest at rates other than 6% may be found as 
follows: 

To find the interest at: 

6£% . . . add T V to the interest at 6%. 

6£% . . . add £ to the interest at 6%. 

7 % . . . add | to the interest at 6%. 

7£% . . . add £ to the interest at 6%. 

7|% . . . add 5 to the interest at 6%. 

8 % . . . add | to the interest at 6%. 

9 % . . . add | to the interest at 6%. 

10 % . . . divide the interest at 6% by 6, and move the decimal point one 

place to the right. 

12 % . . . multiply the interest at 6% by 2. 

5£% . . . deduct T V from the interest at 6%. 

5j% . . . deduct £ from the interest at 6%. 

5 % . . . deduct £ from the interest at 6%. 

4|% . . . deduct f from the interest at 6%. 

4£% . . . deduct £ from the interest at 6%. 

4 % . . . deduct £ from the interest at 6%. 

3 % . . . divide the interest at 6% by 2. 

2 % . . . divide the interest at 6% by 3. 

1 % . . . divide the interest at 6% by 6. 

To find the interest at any other rate, divide the interest at 6% by 6, 
to find the interest at 1%, and then multiply the interest at 1% by the 
given rate. 

ILLUSTRATIONS 


636. 1 . If $16.3875 is the in¬ 
terest at 6%, find the interest at 
«%. 

12)$16.3875 = int. at 6% 
1.3656 = int. at j% 
$15.0219 = int. at 5|% 

3. If the interest at 6% is 
$137.24, find the interest at 
7|%. 

4)$137.24 = int. at 6% 

34.31 = int. at lj% 
$171.55 = int. at 7\% 


2. If $58.63 is interest at 
6%, find the interest at 5%. 

6)$58.63 = int. at 6% 

9.7716 = int. at 1% 
$48.8584 = int. at 5% 

4. If interest at 6% is 
73.4276, find the interest at 
5|%. 

8)$73.4276 = int. at 6% 

9.1784 = int. at |% 
$64.2492 = int. at 5i% 






308 


INTEREST 


5. Find the interest on $548.56 for 42 days at (a) 4j%, and (6) 
at 10%. 

(a) $ 548.56 = int. for 6 days at 6%. (b) 

$3 839.92 = int. for 42 days at 6%. 6 ) $3.83992 = int. at 6%. 

639.98 = int. for 42 days at 1%. $6.3998 = int. at 10%. 

41 X $.63998 = $2.71991, int. 

at 41%. 

638. Find the interest at 6|% and at 7% in Art. 633, problems 
1-12 inclusive; at 1\% and at 4^% in problems 13-24 inclusive; 
at 5|% and at 5% in problems 25-38 inclusive; and at 10% and 
4% in problems 39-52 inclusive. 


SHORT METHODS 

639. What is the interest at 6% on $485 for 60 da.? 

What is the interest at 6% on $485 for 600 da.? 

What is the interest at 6% on $485 for 6000 da.? 

How does the interest for 600 da. compare with the interest for 
60 da.? 

How many places are pointed off in the principal to find the 
interest for 60 da.? to find the interest for 600 da.? 

How does the interest for 6000 da. compare with the interest for 
600 da.? How many places are pointed off to find the interest for 
6000 da.? 


640. Find the interest at 6% on $580 for 600 da.; for 6000 da. 
$58.0 = int. for 600 da. 600da. is 10 times60 da.; hence, the inter- 


$580. = int. for 6000 da. 


est for 600 da. is 10 times the interest for 60 


da. Since the interest for 60 da. is found by 
pointing off two places in the principal, the interest for 600 da. is found by 
pointing off one place (Prin. 1, page 116). 

Because 6000 da. is 10 times 600 da., the interest for 6000 da. is found by 
pointing off no places in the principal. That is, the interest at 6% equals the 
principal in 6000 da. 


641. Summary of principles: 

1. Pointing off 3 places in the principal gives the interest at 6% for 
6 da. 

2. Pointing off 2 places in the principal gives the interest at 6% for 
60 da. 





INTERCHANGING PRINCIPAL AND TIME 


309 


3. Pointing off 1 place in the principal gives the interest at 6% for 

600 da. 

4. Pointing off no places in the principal gives the interest at 6% 
for 6000 da. 


642. Find the interest at 6% on: 
1. $492 for 600 da. 


2. $1486 for 1200 da. 

3. $1590 for 1800 da. 

4. $1840 for 6000 da. 

5. $2460 for 3000 da. 

6. $1230 for 1500 da. 

7. $6280 for 4500 da. 

8. $9684 for 2000 da. 

9. $8346 for 900 da. 

10. $9471 for 200 da. 


11. $1760 for 3600 da. 

12. $1840 for 4200 da. 

13. $1932 for 4800 da. 

14. $750.75 for 6000 da. 

15. $916.73 for 900 da. 

16. $813.25 for 12,000 da. 

17. $532.37 for 6000 da. 

18. $928.50 for 800 da. 

19. $4328 for 300 da. 

20. $7388 for 1500 da. 


INTERCHANGING PRINCIPAL AND TIME 

643. Time is rarely, if ever, expressed in days, except for parts of 
a year. Hence, the problems in the preceding exercise are not of 
direct practical importance. Indirectly the solutions are valuable. 
By the commutative law of multiplication, the interest on $6000 
for 89 da. is the same as the interest on $89 for 6000 da., as shown 
in the following illustration. 


Find the interest at 6% on $6000 for 89 da. 


(a) $60 

00 = int. for 60 da. 

20 

00 = int. for 20 da. 

6 

00 = int. for 6 da. 

3 

00 = int. for 3 da. 

$89 

00 = int. for 89 da. 


( b ) $89 = int. for 6000 da. 


In solution (a) the direct solution is given by finding the interest first for 60 
da., then for 20 da., 6 da., and 3 da., the sum of the several amounts being $89 
for 89 da. 

In solution ( b ) the principal, $6000, and the time, 89 da., are interchanged 
so that the problem would read “Find the interest on $89 for 6000 da.” But 
the interest at 6% on any principal for 6000 da. is equal to the principal. 
Therefore the interest is $89. 




310 


INTEREST 


Proof of the Principle 


For the interest on $1800 for 47 da. at 6%. 


$1800 at 6% int. for 47 da. = $47 at 6% int. for 1800 da. 


$1800 X .06 X 47 
360 


$47 X .06 X 1800 
360 


$14.10. 


By the cancellation method of finding interest the principal multiplied by 
the rate per annum, multiplied by the time expressed as years or a fraction of a 
year, gives the interest. In the illustration here given it is to be observed that 
both the principal and the number of days are above the line,—that is, they 
are each a part of a dividend to be divided by the number below the line. Since 
both principal and time (in days) are factors of the same product, their relative 
order in the list of factors is immaterial (Prin. 13, page 117). Therefore, the 
principal and the time in days can be interchanged without altering the amount 
of the interest. 


644. By interchanging principal and time, find the interest at 


6% on: 


1. $6000 for 29 da. 

19. $60 for 142 da. 

2. $3000 for 47 da. 

20. $90 for 101 da. 

3. $600 for 83 da. 

21. $150 for 140 da. 

4. $1200 for 19 da. 

22. $1500 for 64 da. 

5. $1500 for 37 da. 

23. $2100 for 44 da. 

6. $2000 for 81 da. 

24. $2700 for 14 da. 

7. $12,000 for 43 da. 

25. $6600 for 31 da. 

8. $9000 for 53 da. 

26. $7200 for 25 da. 

9. $8000 for 79 da. 

27. $7800 for 17 da. 

10. $1000 for 33 da. 

28. $8400 for 13 da. 

11. $180 for 55 da. 

29. $540 for 38 da. 

12. $120 for 74 da. 

30. $630 for 41 da. 

13. $300 for 71 da. 

31. $72 for 35 da. 

14. $480 for 92 da. 

32. $30 for 80 da. 

15. $360 for 103 da. 

33. $240 for 70 da. 

16. $200 for 110 da. 

34. $15 for 100 da. 

17. $900 for 111 da. 

35. $15,000 for 160 da. 

18. $600 for 132 da. 

36. $12,600 for 131 da. 




INTERCHANGING PRINCIPAL AND TIME 


311 


645. l. Find the interest on $750 at 6% from July 8, 1924, to 
Sept. 4, 1924. (Exact time.) 


July, 23 da. 
Aug., 31 da. 
Sept., 4 da. 
58 da. 


$5 80 = 600 days’ int. 

145 = 150 days’ int. 
$7 25 = 750 days’ int. 


The exact time is found by 
counting all the days after 
July 8 to and including Sept. 
4. It is 58 da. Interchange 
principal and time; the inter¬ 
est is found to be $7.25. 


2. Find the interest at 6% on $897.50 from May 15, 1924, to 
Aug. 30, 1925. (Compound time.) 


yr. 

mo. 

da. 

$8 

97.50 

= 

int. for 2 

mo. 

1925 

8 

30 

62 

82 5 

= 

int. for 14 

mo. 

1924 

5 

15 

4 

48 75 

= 

int. for 1 

mo. 

1 

3 

15 

2 

22 37 

= 

int. for 15 

da. 




$69 

55 62 





$69.56 

= int. 

for 1 

yr- 

, 3 mo. 15 da. 


The compound time is 1 yr. 3 mo. 15 da.; the interest is found as explained 
in Art. 632. 


3. Find the interest and the amount at 6% on $920 from April 
14, 1924, to Sept. 2, 1924. (Bankers’ time.) 

From April 14 to Aug. 14 is 4 mo. 

From Aug. 14 to Sept. 2 is 19 da. 

Therefore the time is 4 mo. 19 da. 


$9 

20 

= int. for 

2 

mo. 

18 

40 

= int. for 

4 

mo. 

2 

30 

= int. for 

15 

da. 


46 

= int. for 

3 

da. 


1533 

= int. for 

1 

da. 

$21 

3133 




$21. 

.31 = 

int. for 4 mo 

. 19 


$920 + $21.31 = $941.31, 


The time is found by count¬ 
ing the whole months for the 
months and the exact days for 
the fraction of a month. The 
time is 4 mo. 19 da. Having 
the time, the interest is found 
as previously explained. The 
amount is equal to the princi¬ 
pal plus the interest. 


646. Find the interest and the amount at 6%, (1) for compound 
time, (2) for exact time, and (3) for bankers’ time on each of the 
following: 

1. $420 from Apr. 4, 1924, to Jan. 10, 1925. 

2. $640 from Aug. 7, 1924, to Feb. 4, 1925. 

3. $560 from May 8, 1924, to Dec. 1, 1925. 










312 


INTEREST 


4. $727.50 from Mar. 7, 1924, to Nov. 8. 1924. 

5. $643.75 from Jan. 8, 1923, to Oct. 5, 1923. 

6. $798.40 from Feb. 28, 1923, to Sept. 13, 1923. 

7. $1260 from July 1, 1924, to Mar. 14, 1925. 

8. $1375 from May 10, 1924, to Feb. 10, 1925 

9. $1760 from June 1, 1924, to Dec. 30, 1924. 

10. $1840 from Oct. 17, 1924, to July 10, 1925. 

11. $475.80 from Aug. 31, 1924, to May 15, 1925. 

12. $768.50 from Dec. 8, 1924, to June 4, 1925. 

13. $978.30 from Nov. 17, 1924, to Mar. 24, 1925. 

14. $7485 from Oct. 8, 1924, to Aug. 6, 1925. 

15. $7382 from Aug. 12, 1924, to Jan. 2, 1925. 

16. $6600 from May 15, 1924, to Feb. 4, 1925. 

17. $7500 from June 9, 1924, to Oct. 1, 1924. 

18. $8000 from Mar. 1, 1924, to Dec. 10, 1924. 

19. $12,000 from Feb. 14, 1922, to Oct. 1, 1923. 

20. $1000 from Aug. 13, 1924, to June 20, 1925. 

21. $900 from Dec. 18, 1924, to July 30, 1925. 

22. $770 from Oct. 15, 1924, to Jan. 2, 1926. 

23. $990 from May 16, 1924, to Nov. 24, 1925. * 

24. $650 from Jan. 22, 1924, to Dec. 13, 1924. 

25. $1860 from Feb. 1, 1924, to Jan. 8, 1925. 

26. $3600 from Mar. 4, 1924, to Oct. 17, 1925. 

27. $4200 from June 30, 1924, to Jan. 1, 1925. 

28. $426.74 from May 8, 1924, to Dec. 4, 1924. 

29. $587.50 from Mar. 4, 1924, to Dec. 25, 1924. 

30. $736.40 from Aug. 17, 1924, to July 1, 1925. 

31-40. Find the interest and the amount at 5% and 7% on 
problems 1-10 inclusive. 

41-50. Find the interest and the amount at 5J% and at 6§% on 
problems 11-20 inclusive. 

51-60. Find the interest and the amount at 1\% and at 4% on 
problems 21-30 inclusive. 


ACCURATE INTEREST 


313 


ACCURATE INTEREST 


647. Accurate interest is computed for exact time in days, and on 
a basis of 365 da. to the year. It is used chiefly by the United 
States government, and by some bankers. Business men and 
bankers generally find the computation of accurate interest too 
inconvenient, though some merchants and bankers use it with 
the help of interest tables.* 

648. Since by the “ordinary” interest methods, 360 da. are 
counted as a year, 360 days’ interest by the ordinary method is 
greater than 360 days’ interest by the exact interest method, by as 
much as -§§-& is greater than , or . That is, for any given 
number of days, the ordinary interest is greater than the 
accurate interest at the same rate per cent. Hence, to change 
accurate interest to ordinary interest, add ^ of the accurate in¬ 
terest to itself; and to change ordinary interest to accurate interest, 
deduct of the ordinary interest from itself. 

649. In practice, however, accurate interest is computed as 
shown in the following illustrative example: 

l. Find the accurate interest on $790 at 5% for 140 da. 

28 

$790 X .05 X U0 1106 

= $15.15, the accurate interest. 


m 

73 


73 


Multiplying $790 by 5% gives the interest for one year. Dividing that 
product by 365 gives the interest for 1 da., which multiplied by 140 gives the 
interest for 140 da. 

650. Find the accurate interest on: 


1. $750 for 148 da. at 6%. 

2. $675 for 213 da. at 4%. 

3. $876 for 219 da. at 5%. 

4. $1314 for 58 da. at 7%. 

5. $1825 for 64 da. at 6^%. 

11. $4564 from Aug. 11, 1924, to May 15, 1925, at 5%. 

12. $3870 from June 1, 1924, to Dec. 30, 1925, at 6%. 

* Bankers and others use interest tables for both ordinary and accurate 
interest. The tables are too long to permit of illustration in this book. 


6. $1600 for 13 da. at 4£%. 

7. $1500 for 18 da. at 3%. 

8. $1700 for 38 da. at 5%. 

9. $4380 for 51 da. at 6%. 
10. $3825 for 47 da. at 5%. 




314 


INTEREST 


13. $4330 from Jan. 3, 1923, to Oct. 14, 1923, at 9%. 

14. $5786 from Feb. 23, 1924, to Jan. 4, 1925, at 6%. 

15. $3342 from May 16, 1924, to Aug. 1, 1924, at 10%. 

16. $5540 from Oct. 21, 1924, to Mar. 1, 1925, at 12%. 

17. $48.75 from Jan. 5, 1922, to Aug. 4, 1926, at 6%. 

18. $125.50 from Dec. 3, 1924, to Nov. 15, 1926, at 5%. 

19. $456.73 from Oct. 18, 1924, to Jan. 31, 1925, at 4%. 

20 . $592.15 from June 11, 1924, to Dec. 7, 1924, at 6%. 

21 . $678.25 from Sept. 2Q, 1924, to Sept. 13, 1925, at 5%. 

22 . $431.25 from Apr. 19, 1924, to Aug. 10, 1925, at 6%. 

23. $598.80 from Mar. 31, 1923, to May 1, 1926, at 6%. 

24. $163.25 from July 3, 1924, to Dec. 24, 1925, at 5%. 


NEGOTIABLE PAPER 


651. Negotiable paper is business paper which may be trans¬ 
ferred by indorsement and delivery, or by delivery only, the trans¬ 
feree taking good title thereto. The principal forms of negotiable 
paper are checks, notes, drafts, and certificates of deposit. 

652. A check is a written order by a depositor on a bank 0 r 
banker payable on demand. 


653. 


Form op a Check 



Hbc (RmsterdarrvBanb 



654. In this check S. B. Kooper is the drawer; Walter G. Lind¬ 
say is the payee; and the bank is the drawee. 



















NEGOTIABLE PAPER 


315 


655. 


Form of a Bank Draft 


^hel^run^vpickl&anlo 





*" £m rss££2“‘' '-e*&L.-*& r 


656. A bank draft is a check drawn by one bank on another 
bank. In the above form the Brunswick Bank is the drawer; 
James E. Winter is the payee; and the Mercantile National Bank 
is the drawee. 

657. A draft is a written order by one party on another directing 
the payment of a specified sum of money at a certain time to a per¬ 
son named therein, to order or to bearer. 

658. In the following draft, F. G. Huxley is the drawer. He is 
the person who wrote or “drew” the draft. He has addressed the 
draft to C. D. Evarts, who is called the drawee. The drawee is 
the person drawn on,—the one who is directed to pay. L. M. 
Norton is the payee. Huxley wrote the draft and gave it to 
Norton, who will present it to Evarts for payment when it is due. 


























316 


INTEREST 


660. Form of a Time Draft (After Sight) 



It should be observed that in the first of the preceding draft 
forms, the time is “Sixty days after date,” and in the second it is 
“Thirty days after sight” In the first form the time is reckoned 
from the date of the draft, Aug. 30,19 , while in the second form 

it is reckoned from the date of acceptance, June 5,19 . 

661. Acceptance is the written promise (an oral acceptance is 
legal and binding) of the drawee to pay the draft when it is due. 
Since the time (in drafts of the second form) is reckoned from the 
time of acceptance, these drafts must be accepted,’and the accept-* 
ance must be dated. Drafts of the first form do not require 
acceptance. 

662. Until the drawee accepts a draft, he is in no way liable for 
its payment. When he accepts it he becomes liable for its pay¬ 
ment the same as the maker of a note. 

663. Form of a Sight Draft 





























NEGOTIABLE PAPER 


317 


664. A sight draft is payable on demand, the expression “at 
sight” meaning when the draft is presented by the payee to the 
drawee, for payment. 

665. 

Form of a Trade Acceptance 




Trade Acceptance 

NewYpi-ft 2-S , .19^1 z. 





The 


T° 




p A y to the order of OtirSClVeS 


a q/jLc. 

acceptor hereof arises4ut of the purchase 
of goods from the drawer 


DUC .YZl2-TA^--2-f: > _19_ 


>,?— 


A trade acceptance is a time draft or bill of exchange, drawn 
by the seller of goods on the buyer for the purchase price, and 
accepted by the buyer, payable on a certain date at a place 
designated on the face of the instrument. 


666 . A trade acceptance differs from an ordinary draft in that 
its use is restricted to the one purpose of drawing on the buyer 
of merchandise for the selling price. It is a two-name paper, 
while a draft may be a three-name paper. A draft is used for a 
variety of purposes. 


667. A bank acceptance is a draft or bill of exchange, of which 
the acceptor is a bank or trust company, or a person or company 
engaged generally in the business of granting bank acceptance 
credits. In form it frequently does not differ from the Trade 
Acceptance. 












318 


INTEREST 


668 . A bank acceptance is used when the buyer of goods wishes 
to use a bank’s credit instead of his own credit. The bank, for a 
small fee, agrees to accept a draft or drafts drawn upon it covering 
certain agreed purchases by the buyer. The bank gives the 
buyer a written agreement to accept drafts for him, which the 
buyer sends to the seller. On the strength of this promise the 
seller ships the goods, and draws on the bank for the selling price. 

669. The seller of the goods attaches to the draft the bill of 
lading to show that the goods are shipped, and either sells the 
draft to his bank or asks his bank to collect the amount for him. 
The draft is then forwarded to the accepting bank for acceptance 
and payment. 

670. A promissory note is a written promise to pay to a person 
named in the note, to order or to bearer, a specified sum of money. 


671. Form op a Promissory Note 



672. In the above note S. L. Newhouse is the maker, and F. J. 
Gildersleeve is the 'payee. 

673. A certificate of deposit is a written statement issued by a 
bank to a person who deposits money for safe keeping. It contains 














NEGOTIABLE PAPER 


319 


the promise of the bank to return the specified sum on the return 
of the certificate. 

674. Form of a Certificate of Deposit 


VimmMKhmw ttotk 



675. Indorsement is anything written on the back of commer a 
paper, which has reference to the paper itself. Indorsement is for 
one of three purposes. It may be for the purpose: (1) of transfer¬ 
ring the paper; (2) of adding to its security; or (3) of recording a 
partial payment of the paper. 

For the transfer of paper, indorsements are of several kinds, the 
most important of which are: 

(1) Blank Indorsement, written, Robert C. Ogden. 

It consists of only the name of the indorser. 

(2) Special Indorsement, written, Pay to the order of 

J. J. Astor, 

A. T. Stewart. 

It directs payments to some specified person, to order, or to 
bearer. 

(3) Qualified Indorsement, written, 

Without Recourse. Pay to the order of 

W. C. Mooney or W. C. Mooney 

without recourse. 

F. R. Crook 

The Wiords “without recourse” have the effect of excusing the 
indorser from further liability of payment. 















320 


INTEREST 


676. When the indorsement is for the purpose of adding to the 
security of the paper, it is generally written as a blank indorsement. 

677. Each indorser, except those who use the qualified indorse¬ 
ment, is liable for the payment of the paper if the party primarily 
liable fails to pay. 

678. An indorsement of a partial payment is a record of the 
amount and date of the payment. Thus: 

Received May 4, 1924, $500 
Received Dec. 1, 1924, $600 
Received Oct. 1, 1925, $1000 

679. If negotiable paper is not accepted when presented for 
acceptance, or is not paid at maturity, it is said to be dishonored. 

680. Dishonored negotiable paper should be protested. Protest 
is the official act of a notary public in certifying to the dishononof a 
negotiable instrument. 

681. A notice of protest is sent immediately by the notary public 
to all parties conditionally liable for payment of the paper. 

682. In case of failure to give notice of protest, all parties con¬ 
ditionally liable for payment are excused from their liability. 

BANK DISCOUNT 

683. Bank discount is a deduction made from the amount due at 
maturity on notes and drafts in consideration of their being cashed 
or bought before maturity. 

684. If a merchant has received from one of his debtors a draft 
payable in 60 da., and does not wish to wait for payment till it 
matures, he can take it to a bank and sell it, or discount it, as it is 
called. The discount is determined by reckoning the simple in¬ 
terest on the amount of the draft from the day it is discounted to 
the day of maturity. The merchant will receive the proceeds, 
which is the difference between the amount due on the draft at 
maturity and the discount. 

685. As a rule banks charge interest, or discount, for the exact 
number of days from the day of discount to the day of maturity. 
This time is called the term of discount. 


BANK DISCOUNT 


321 


Notes. 1 . In a few states, viz. Ky., La., Md., Pa., and Va., the day of 
discount as well as the day of maturity is included in the term of discount, thus 
adding one day to the time. 

2. In Ariz., Calif., Ga., Ky., Miss., S. Dak., Tex., and Wis., notes, ac¬ 
ceptances, etc., falling due on Saturday are due and payable on the same day. 
In all other states they are due and payable the next succeeding business day. 

3. In all states, notes, acceptances, etc., falling due on a holiday or on 
Sunday, are due and payable the next succeeding business day. 

4. In discounting paper falling due on Saturday, Sunday, or a legal holiday, 
banks charge interest and discount until the “next succeeding business day” 
(except where payable the same day). 

5. Answers to problems in bank discount in this book are prepared in 
accordance with the most common method used by bankers. It is suggested 
that the student follow the custom of his own state. 

686. Days of grace are tliree days allowed by law, for the pay¬ 
ment of notes and drafts, beyond the expiration of the time 
specified in the paper. The practice of allowing three days of 
grace, which was formerly very common, has now been abolished 
in most states. 

687. The solution of problems in bank discount is based on the 
principles of simple interest, or of percentage. 

The value of the note at maturity corresponds to the 'principal in 
interest, or to the base in percentage . 

The rate of discount corresponds to the rate per cent in interest 
and in percentage. 

The bank discount corresponds to the interest or the percentage 
and proceeds to the difference. 

688. To find the date of maturity and the term of discount. 

1 . Find the date* of maturity and the term of discount of a 3 
months' note dated Apr. 14, and discounted May 3. 

Apr. 14 + 3 mo. = July 14, date of maturity. 

From May 3 to July 14 = 72 da., term of discount. 

The date of maturity is found by counting forward 3 mo. from Apr. 14, which 
gives July 14. The term of discount is found by counting all the days from the 
date of discount, May 3, to the date of maturity, July 14. There are 28 da. 
remaining in May, 30 da. in June, and 14 da. in July, the sum of which is 
72 da. (See Art. 449 page 194.) 

VAN TUYL’S NEW COMP. AR.—21 


322 


INTEREST 


2. Find the date of maturity and the term of discount of a 60 
days' sight draft, dated Aug. 18, accepted Aug. 21, and discounted 
Sept. 4. 

Aug. 21 + 60 da. = Oct. 20, date of maturity. 

From Sept. 4 to Oct. 20 = 46 da., term of discount. 

A draft drawn at 60 days’ sight* does not mature until 60 da. after it is 
accepted, hence, the day of maturity in this draft is 60 da. after Aug. 21, which 
is Oct. 20. Exact days must be counted when the time is expressed in days. 
(See Art. 455, page 197.) The term of discount is the exact number of days 
from Sept. 4 to Oct. 20, which is 46 da. 


689 . Copy the following and fill in the dates of maturity and the 
terms of discount: 



Date of 
Paper 

Time 

Date \>f 
Acceptance 

Due 

Date 

Date of 
Discount 

Term of 
Discount 

1. 

Jan. 15 

4 months 

f 


Mar. 10 


2. 

Feb. 23 

90 days 



Mar. 1 


3. 

June 10 

3 months 



July 18 


4. 

May 8 

90 days after sight 

May 14 


May 26 


5. 

Oct. 7 

2 months 



Oct. 21 


6. 

Aug. 11 

30 days after sight 

Sept. 1 


Sept. 10 


7. 

July 8 

60 days 



Aug. 14 


8. 

May 1 

45 days 



May 14 


9. 

Apr. 3 

2 months after date 

May 1 


May 2 


10. 

Feb. 18 

90 days 



Mar. 31 


11. 

June 17 

3 months 



June 24 


12. 

July 27 

60 days’ sight 

Aug. 4 


Aug. 7 


13. 

Sept. 19 

4 months 



Nov. 10 


14. 

Mar. 7 

1 month 



Mar. 9 


15. 

Apr. 13 

60 days after date 

May 1 


May 4 


16. 

Jan.28 

3 months after sight 

Feb. 1 


Mar. 8 


17. 

June 15 

5 months 



Sept. 30 


18. 

Aug. 11 

90 days 



Sept. 30 


19. 

May 21 

40 days’ sight 

June 1 


June 10 


20. 

June 16 

2 months 



July 1 


21. 

Dec. 13 

3 months 



Feb. 7 


22. 

Nov. 30 

3 months 



Jan. 2 


23. 

Nov. 30 

90 days 



Dec. 1 


24. 

Oct. 9 

90 days after date 

Oct. 12 


Nov. 2 


25. 

Mar. 14 

30 days’ sight 

Mar. 20 


Mar. 24 



* See Art. 660, page 316. t Blank spaces in this column are not to be filled in. 













BANK DISCOUNT 


323 


690. To find the bank discount and the proceeds of a note or 
draft. 


Find the date of maturity, the term of discount, the bank dis¬ 
count, and the proceeds of a 90-da. note of $1500, dated May 1 
and discounted May 16, at 6%. 

May 1 + 90 days = July 30, date of maturity. 

From May 16 to July 30 = 75 days, term of discount. 


$15 

00 

3 

75 

$18 

75 


75 = discount for 75 days. 
$1500 — $18.75 = $1481.25, proceeds. 


The date of maturity and the terms of discount are found as in the preceding 
exercise. The bank discount is the simple interest on the value of the note for 
the term of discount, 75 days, which is $18.75. The proceeds is the difference 
between the bank discount and the value of the note;$1500—$18.75 = $1481.25. 


691. Find the date of maturity, the term of discount, the bank 
discount, and the proceeds of the following notes and drafts: 



Face 

Time 

Date op 
Paper 

Date of 
Acceptance 

Date of 
Discount 

Rate of 
Discount 

l. 

$1800.00 

4 months 

Aug. 1 


Aug. 11 

6% 

2. 

2000.00 

3 months 

July 11 


Aug. 1 

6% 

3. 

1640.00 

90 days 

May 13 


June 4 

6% 

4. 

1150.00 

60 days after sight 

Mar. 18 

Mar. 24 

Apr. 7 

5% 

5. 

1990.00 

30 days’ sight 

Apr. 27 

May 3 

May 6 

5% 

6. 

2870.00 

5 months 

June 19 


Aug. 1 

6% 

7. 

3675.00 

2 months 

July 17 


Aug. 1 

7% 

8. 

9400.00 

60 days’ sight 

Oct. 11 

Oct. 18 

Nov. 1 

6% 

9. 

8478.00 

30 days’ sight 

May 28 

June 4 

June 11 

41% 

10. 

7200.00 

3 months 

Aug. 13 


Aug. 19 

4% 

11. 

5900.00 

1 month 

Jan. 31 


Feb. 4 

51% 

12. 

8212.00 

2 months 

July 1 


Aug. 14 

6% 

13. 

4750.00 

4 months 

Oct. 6 


Nov. 6 

71% 

14. 

5125.00 

6 months 

June 30 


Oct. 18 

6% 

15. 

6000.00 

90 days after date 

Sept. 5 

Sept. 25 

Oct. 11 

6% 

16. 

5400.00 

60 days after sight 

Aug. 4 

Aug. 11 

Aug. 23 

8% 

17. 

4500.00 

30 days’ sight 

July 5 

July 11 

July 11 

71% 

18. 

3100.00 

5 mo. after date 

Jan. 12 

Mar. 11 

Mar. 13 

6% 

19. 

412.50 

3 months 

Apr. 12 


May 1 

9% 

20. 

368.75 

90 days 

May 10 


June 11 

10% 

















324 


INTEREST 


692. Bankers often use a table like the following to find the 
number of days between two dates: 

Bankers’ Time Table 


To the Same Day of the Next 


From any 


Day of 

Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

Jan. 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

Feb. 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

Mar. 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

Apr. 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

June 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

July 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

Aug. 

153 

184 

212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

Sept. 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

Oct. 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

Nov. 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

Dec. 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 


693. How to use the table. 

The table gives at a glance the exact number of days from any day of any 
month to the corresponding day of any other month for periods of time not 
greater than one year. Thus, the number of days from the 18th of May to the 
18th of the next January is found by taking the number to the right of the 
month of May in the “January column,” which is 245 da. From the 9th of 
April to the 9th of December is 244 da., found on the “April line” and the 
December column. 

From Mar. 15 to July 28 is found thus: Take from the table the number of 
days from Mar. 15 to July 15, which is 122 da., and add to it the number of 
days from July 15 to July 28. From July 15 to July 28 is 13 da., which, added 
to 122 da., equals 135 da. 

For the time from Oct. 25 to Apr. 10, find the time from Oct. 25 to Apr. 25 
which is 182 da., and deduct the number of days from Apr. 10 to Apr. 25, which 
is 15 da. 182 da. less 15 da. leaves 167 da., the time from Oct. 25 to Apr. 10. 

694. Find, by using the table, the number of days from: 

1. Aug. 4 to Mar. 4. 4. Mar. 3 to Jan. 3. 

2. Jan. 8 to Dec. 8. 5. May 30 to Nov. 30 

3. Apr. 11 to Aug. 11 6. July 1 to Jan. 1. 






















BANK DISCOUNT 


325 


695. To find the proceeds of a note or draft when collection is 
charged. 

Find the proceeds of a 90-da. note of $3800 dated June 3, 1924. 
if discounted July 1, 1924, at 6%; collection yu%. 

June 3 + 90 da. = Sept. 1, due date. 

From July 1 to Sept. 1 = 62 da., term of discount. 

$38 00 = discount for 60 da. 

127 = discount for 2 da. 

$39 27 = discount for 62 da. 
ro% of $3800 = $3.80, collection. 

$39.27 + $3.80 = $43.07, the bank's total charge. 

$3800 — $43.07 = $3756.93, the proceeds. 

The collection charge is reckoned on the value of the note; T V% of $3800 is 
$3.80. The bank’s total charge is the sum of the bank discount,. $39.27, and 
the collection charge, $3.80, or $43.07. $3800 — $43.07 = $3756.93, proceeds. 


696. To find the proceeds of an interest-bearing note. 

Find the proceeds of a 4 months' 6% interest-bearing note of 
$3200, dated May 15, 1924, and discounted June 10, 1924 at 6%; 
collection \%. 

May 15 + 4 mo. = Sept. 15, due date. 

From June 10 to Sept. 15 = 97 da., term of discount. 

$32j00 = int. for 2 mo. $3200 + $64 = $3264, value of the 
64[00 = int. for 4 mo. note of maturity. 

= dis. for 60 da. \% of $3264 = $4.08, collection. 

= dis. for 30 da. $52.77 + $4.08 = $56.85, bank's 

= dis. for 6 da. total charge. 

= dis. for 1 da. $3264 — $56.85 = $3207.15, pro- 
= dis for 97 da. ceeds. 


32 

64 

16 

32 

3 

264 


544 

$52 

768 


The due date and the term of discount are first found, as explained on page 
321, Art. 688. Next the interest on the face of the note, $3200, is found for the 
entire time the note has to run, 4 mo. Then the interest, $64, is added to the 
face of the note, $3200, which gives $3264, the value of the note at maturity. 
The discount is then reckoned for 97 da. on the full value of the note, and 
amounts to $52.77. The collection charge is reckoned on the value of the 
note at maturity, $3264. \% of $3264 equals $4.08. The bank’s total charge 

is equal to the sum of the bank discount and the collection, which is $56.85. 
$3264 — $56.85 = $3207.15, proceeds of note. 




326 


INTEREST 




697 . Discount Register 


No. 

For Whom Discounted 

Rate 

Indorser 

Collateral 

Date 

Time 

1. 

A. C. Montgomery 

6% 



Jan.10 

4 mo. 

2. 

D. C. Tarbell . . 

6% 

A. G. Ward 


Mar. 15 

3 mo. 

3. 

M. L. Hess . . . 

6% 

W. C. Dowd 


June 8 

90 da. 

4. 

G. H. Dexter . . 

6% 


$2500 U.S. 3s 

Oct. 17 

60 da. 

5. 

S. M. Hamilton . 

6% 

G. 0. Hall 


Aug. 12 

60 da. 

6. 

J. S. Lyman . . 

5% 



Oct. 11 

6 mo. 

7. 

L. L. Holmes . . 

4% 



June 12 

90 da. 

8. 

A. H. Beeman . . 

4J% 


$5000 N.Y.C. 4s 

Apr. 20 

75 da. 

9. 

L. S. Pierce . . . 

6% 

A. Sohmer 


May 10 

40 da. 


Above and on the opposite page is shown a form of a bank 
discount register. Rule a similar form and fill in the missing 
items. 


698 . Find the proceeds of the following notes: 



Face 

Time 

Date 

Interest 

Date of 
Discount 

Rate of 
Discount 

Collec¬ 

tion 

1. 

$1400 

2 mo. 

Aug. 1 

6% 

Aug. 10 

6% 

Wo 

2. 

1550 

3 mo. 

May 15 

5% 

June 1 

5% 


3. 

2160 

90 da. 

Feb.15 

8% 

Mar. 17 

7% 

Wo 

4. 

560 

60 da. 

Dec. 10 

6% 

Dec. 20 

8% 

Wo 

5. 

940 

5 mo. 

July 1 

6% 

Sept. 10 

5% 

Wo 

6. 

1120 

4 mo. 

Oct. 20 

5% 

Dec. 1. 

6% 


7. 

4382.75 

90 da. 

Nov. 17 


Nov. 29 

6% 

tV% 

8. 

526.80 

2 mo. 

Dec. 8 


Dec. 13 

5% 

Wo 

9. 

2600 

3 mo. 

Mar. 8 


Mar. 13 

6% 


10. 

557.68 

2 mo. 

Dec. 4 


Dec. 13 

5% 

Wo 

11. 

943.75 

4 mo. 

Oct. 1 


Oct. 2 

6% 


12. 

321.60 

30 da. 

Jan.31 


Feb. 4 

n% 

tV% 

13. 

416.50 

60 da. 

Mar. 31 


Apr. 13 

4*% 

Wo 

14. 

531.60 

3 mo. 

Nov. 30 


Jan. 3 

5% 

Wo 

































BANK DISCOUNT 


327 


Discount Register 


Face 

Rate 
of Int. 

Int. 

Amount 

Due 

Date 
of Dis. 

Da. 

Dis. 

Rate of 
Exch. 

Exch. 

Proceeds 

$5000 





March 8 



■Jf hs% 



$4500 





Apr. 13 



tV% 



$2500 





July 1 



i% 



$1600 





Nov. 10 



1% 



$1748 





Sept. 1 



— 



$1372 





Nov. 4 



i% 



$2875 

6 % 




June 27 



*% 



$3125 

n% 




May 5 



i% 



$4700 

8 % 




May 14 



— 




15. 


$4728.40 Rochester, N. Y., June 3, 19 

Two months after date we promise to pay to the order of- 

_C. J. Hermans & Co____ 

Forty-seven hundred twenty-eight:-- tVtt Dollars, 

with interest at 6%. Value received. 

F. F. Jones & Co. 


Discounted June 8 at 5%; collection £%. 



Discounted Jan. 13 at 6%; collection \%. 






































328 


INTEREST 


PRESENT WORTH AND TRUE DISCOUNT 

699. How much will $1 amount to in 1 yr. at 6%? 

At the same rate, how many dollars will, in 1 yr., amount to 
$3.18? $5.30? $10.60? $21.20? $212? $848? 

How much will $1 amount to in 2 mo. at 6%? 

At the same rate, how many dollars will, in 2 mo., amount to 
$2.02? $5.05? $10.10? $808? $2020? $5050? 

700. A man owes a debt of $2550 due in 4 mo. How many 
dollars should he pay his creditor to-day, so that if put at interest 
at 6%, the payment will amount to the face of the debt at 
maturity? 

$1.00 + $.02 = $1.02, amount of $1 at 6% for 4 mo. 

$2550 -I- $1.02 = $2500, amount of payment, or present worth of 
the debt. 

If $1 is placed at interest for 4 mo. at 6%, the interest will be $.02, or 
the dollar will amount to $1.02. Since $1 in 4 mo. at 6% amounts to $1.02, 
it will take as many dollars to amount to $2550 as $1.02 is contained times in 
$2550, or 2500 times. That is, a payment of $2500 placed at interest for 
4 mo. at 6% will amount to $2550. 

701 . The present worth of a debt is a sum which, if put at inter¬ 
est, will amount to the value of the debt at maturity. 

In the above example $2500 is the present worth of the debt because $2500 
placed at interest for 4 mo. will amount to $2550, the value of the debt when it 
is due. 

702. The true discount is the difference between the present 
worth of a debt and the amount due at maturity. 

In the solution just given, the true discount is $50. 

The true discount is equal to the interest on the present worth 
of the debt for the time for which the debt is discounted. 


PROBLEMS 

703. Find the present worth and true discount of each of the 
following debts: 


Debt 

Time Rate op Dis. 

Debt. 

Time 

Rate of Dis. 

1. $520 

8 mo. 

6% 

3. $824 

4 mo. 

9% 

2. $618 

6 mo. 

6% 

4. $1654 

7 mo. 

6% 


PRESENT WORTH AND TRUE DISCOUNT 


329 


5. 

$1212 

3 mo. 

4% 

8. $4100 

10 mo. 

3% 

6. 

$1020 

4 mo. 

6% 

9. $972 

1 yr. 4 mo. 

6% 

7. 

$1260 

1 yr. 

5% 

10. $1100 

2 yr. 6 mo. 

4% 


11. What payment today will cancel a debt of $840 due in 10 
mo. if money is worth 6%? 

12. Find the true discount on a debt of $5100 if paid 4 mo. 
before maturity, interest 6%. 

13. A merchant sells an article for $52.50 on 4 mo. credit. At 
what cash price shall he sell the same article if, to him, money is 
worth 15%? 

14. Find the difference between the simple interest and the true 
discount on $1000 for 6 mo. at 6%. 

15. A merchant’s cash price of a given article is $75, and his 
price at 3 mo. credit is $80. At 6%, how much better is the cash 
price for the buyer? 

16. If a merchant’s 4 months’ credit price of a piano is $420, and 
his cash price is $400, at what rate per annum does he reckon 
money is worth to him? 

17. I am offered a bill of goods invoiced at $1500 on 4 months’ 
credit. In payment I give my note with interest at 6% for a sum 
which, at maturity, will cancel the debt. Find the face of the note. 

18. Compare the true discount with the simple interest on $4080 
for 4 mo. at 6%. How does the difference between the true 
discount and the interest compare with the interest on the true 
discount for the given time and rate? Why? 

19. Find the present worth of a note of $800 drawn at 4 mo. 
from Aug. 1, if discounted Sept. 15, at 7%. (Exact time.) 

20. Apr. 10, I bought a bill of merchandise valued at $5600 on 
5 months’ credit. June 20, I paid $3000 on account. Aug. 1, I 
paid the present worth of the balance of the debt. Reckoning 
interest at 6%, what was the amount of my final payment? 
(Exact time.) 

21. A father wishes to provide a fund of $10,000 for his son 
when he shall become 21 yr. of age. What sum should he invest 
at 5% simple interest on the son’s fifteenth birthday, that it may 
amount to the $10,000? 


330 


INTEREST 


PROBLEMS IN INTEREST 

704. 1 . In what time will any sum of money double itself at 
5%? at 6%? at 8%? at 9%? 

2. At what rate per cent of interest will any sum of money 
double itself in 25 yr.? in 14f- yr.? in 10 yr.? in 8 yr. 4 mo.? 

3. What is the difference between the ordinary interest and the 
accurate interest at 6% on $10,000 for 144 da. 

4. A merchant’s cash price of an article is $35; his price at 4 
months’ credit is $37.50. If money is worth 6% per annum, how 
much has he added to cover cost of collection and bad debts? 

5. A banker pays 3% interest on a deposit of $45,000. He loans 
it 3 mo. at 6%; 4 mo. at 5|%; and for 48 da. at 7%. If it lies idle 
the remainder of the year, does he gain or lose, and how much? 

6. Aug. 1, 1924, a wholesaler sold a bill <5f goods amounting to 
$1875.28, terms net 30 da. If the bill was not paid until Jan. 8, 
1925, what amount was due, interest being 6%? (Exact time.) 

7. On July 8, 1924, a man borrowed money at 6% interest and 
bought 3000 bu. of wheat at $1.08 a bushel. He sold the wheat 
Aug. 17, 1924, at $1.21 a bushel and returned with interest the 
money he borrowed. What was his gain? 

8. A piece of property valued at $8000 is sold on the following 
terms: 20% of the selling price is to be paid in cash; for the re¬ 
mainder five notes falling due in one, two, three, four, and five 
years, respectively, with interest at 6%, are given. If each note 
is paid at maturity, what is the total amount paid for the property? 

9. What amount can a man afford to pay for a house and lot that 
rents for $65 a month so that his investment shall net him 6% 
interest, if he pays for taxes, insurance, repairs, and all other 
expenses, $360 a year? 

10 . A young man had $1600 saved. He purchased a house and 
lot for $4000, borrowing $2400 at a saving bank, giving as security 
a mortgage on the house and lot. The house rented for $37.50 a 
month. He paid 5|% interest on the mortgage; f% premium on 
an insurance policy of $3000; 1.8% tax on a valuation of $3500; 
$14 water rent; and $50 for repairs. What rate of interest did he 
receive on his investment? 


PROBLEMS IN INTEREST 


331 


11. A real estate broker offers me a house and lot for $7500 
guaranteeing an 8% investment. The property is assessed for 
$5000 on which a tax of If % has to be paid. Insurance and other 
expenses will amount to $212.50 a year. At what price must 
the property rent per month to make good the broker’s guarantee? 

12. A merchant bought a bill of goods amounting to $13,875; 
terms 3 mo., 5% cash. He accepted the cash terms. How much 
did he save if money was worth 6%? 

13. An invoice of merchandise was bought on the following 
terms: 4 mo. or 5% cash. The cash discount is equivalent to 
what rate of interest per annum on the gross amount? on the 
net amount? 

14. On an invoice of $4378.60, a merchant is offered 4 mo. 
credit or a discount of 5% for cash. Not having the ready money, 
he accepts the credit terms. What rate per cent of interest does 
he pay on the net amount of the bill? How much would he have 
saved if he had borrowed the money at 6% and paid cash? 

15. Lane and Son bought a bill of merchandise valued at 
$9542.50, terms 3/10, n/60. The bill was dated May 13. Not 
having ready cash to meet the bill May 23, they borrowed at their 
bank on a demand note with interest at 6%, a sum sufficient to 
pay the bill. Assuming that they paid their note at the bank 
the following July 12, find how much they gained by early pay¬ 
ment. 

16. Halsey and Hanson received an invoice of merchandise 
dated July 5, terms 3/10, n/90. The value of the invoice was 
$16,932.75 On July 15, they had only $9000 available cash to 
meet the bill. To raise the rest of the money they wrote their 
note at 80 days from date, for a sum which, when discounted at 
their bank, would yield sufficient proceeds to supplement the 
$9000 to meet the invoice on that day. Find the gain by early 
payment, assuming that money was worth 6% to the firm. 

17. A man on June 7 gave a demand note of $4750 to a bank for a 
loan of that amount. On July 12, he paid the note with interest 
by writing a check for $4773.09. What rate of interest did the 
bank charge him ? 


332 


INTEREST 


705. COMPOUND INTEREST 

Table showing Amount of $1 at Compound Interest in any 


NUMBER OF YEARS, NOT EXCEEDING TWENTY-FIVE 


Pd. 

2 per cent 

2 j per cent 

3 per cent 

3 5 per cent 

4 per cent 

4j per cent 

1 

1.0200 0000 

1.0250 0000 

1.0300 0000 

1.0350 0000 

1.0400 0000 

1.0450 0000 

2 

1.0404 0000 

1.0506 2500 

1.0609 0000 

1.0712 2500 

1.0816 0000 

1.0920 2500 

3 

1.0612 0800 

1.0768 9062 

1.0927 2700 

1.1087 1787 

1.1248 6400 

1.1411 6612 

4 

1.0824 3216 

1.1038 1289 

1.1255 0881 

1.1475 2300 

1.1698 5856 

1.1925 1860 

5 

1.1040 8080 

1.1314 0821 

1.1592 7407 

1.1876 8631 

1.2166 5290 

1.2461 8194 

6 

1.1261 6242 

1.1596 9342 

1.1940 5230 

1.2292 5533 

1.2653 1902 

1.3022 6012 

7 

1.1486 8567 

1.1886 8575 

1.2298 7387 

1.2722 7926 

1.3159 3178 

1.3608 6183 

8 

1.1716 5938 

1.2184 0290 

1.2667 7008 

1.3168 0904 

1.3685 6905 

1.4221 0061 

9 

1.1950 9257 

1.2488 6297 

1.3047 7318 

1.3628 9735 

1.4233 1181 

1.4860 9514 

10 

1.2189 9442 

1.2800 8454 

1.3439 1638 

1.4105 9876 

1.4802 4428 

1.5529 6942 

11 

1.2433 7431 

1.3120 8666 

1.3842 3387 

1.4599 6972 

1.5394 5406 

1.6228 5305 

12 

1.2682 4179 

1.3448 8882 

1.4257 6089 

1.5110 6866 

1.6010 3222 

1.6958 8143 

13 

1.2936 0663 

1.3785 1104 

1.4685 3371 

1.5639 5606 

1.6650 7351 

1.7721 9610 

14 

1.3194 7876 

1.4129 7382 

1.5125 8972 

1.6186 9452 

1.7316 7645 

1.8519 4492 

15 

1.3458 6834 

1.4482 9817 

1.5579 6742 

1.6753 4883 

1.8009 4351 

1.9352 8244 

16 

1.3727 8570 

1.4845 0562 

1.6047 0644 

1.7339 8604 

1.8729 8125 

2.0223 7015 

17 

1.4002 4142 

1.5216 1826 

1.6528 4763 

1.7946 7555 

1.9479 0050 

2.1133 7681 

18 

1.4282 4625 

1.5596 5872 

1.7024 3306 

1.8574 8920 

2.0258 1652 

2.2084 7877 

19 

1.4568 1117 

1.5986 5019 

1.7535 0605 

1.9225 0132 

2.1068 4918 

2.3078 6031 

20 

1.4859 4740 

1.6386 1644 

1.8061 1123 

1.9897 8886 

2.1911 2314 

2.4117 1402 

21 

1.5156 6634 

1.6795 8185 

1.8602 9457 . 

2.0594 3147 

2.2787 6807 

2.5202 4116 

22 

1.5459 7967 

1.7215 7140 

1.9161 0341 

2.1315 1158 

2.3699 1879 

2.6336 5201 

23 

1.5768 9926 

1.7646 1068 

1.9735 8651 

2.2061 1448 

2.4647 1555 

2.7521 6635 

24 

1.6084 3725 

1.8087 2595 

2.0327 9411 

2.2833 2849 

2.5633 0417 

2.8760 1383 

25 

1.6406 0599 

1.8539 4410 

2.0937 7793 

2.3632 4498 

2.6658 3633 

3.0054 3446 


Subtract $1 from the amount in this table to find the interest. 


Amount of SI at Compound Interest in any Number of 


Years not Exceeding Twenty-five 


Pd. 

5 per cent 

6 per cent 

7 per cent 

8 per cent 

9 per cent 

10 per cent 

1 

1.0500 000 

1.0600 000 

1.0700 000 

1.0800 000 

1.0900 000 

1.1000 000 

2 

1 .1025 000 

1.1236 000 

1.1449 000 

1.1664 000 

1.1881 000 

1.2100 000 

3 

1 .1576 250 

1.1910 160 

1.2250 430 

1.2597 120 

1.2950 290 

1.3310 000 

4 

1.2155 063 

1.2624 770 

1.3107 960 

1.3604 890 

1.4115 816 

1.4641 000 

5 

1.2762 816 

1.3382 256 

1.4025 517 

1.4693 281 

1.5386 240 

1.6105 100 

6 

1.3400 956 

1.4185 191 

1.5007 304 

1.5868 743 

1.6771 001 

1.7715 610 

7 

1.4071 004 

1.5036 303 

1.6057 815 

1.7138 243 

1.8280 391 

1.9487 171 

8 

1.4774 554 

1.5938 481 

1.7181 862 

1.8509 302 

1.9925 626 

2.1435 888 v 

9 

1.5513 282 

1.6894 790 

1.8384 792 

1.9990 046 

2.1718 933 

2.3579 477 

10 

1.6288 946 

1.7908 477 

1.9671 514 

2.1589 250 

2.3673 637 

2.5937 425 

11 

1.7103 394 

1.8982 986 

2.1048 520 

2.3316 390 

2.5804 264 

2.8531 167 

12 

1.7958 563 

2.0121 965 

2.2521 916 

2.5181 701 

2.8126 648 

3.1384 284 

13 

1.8856 491 

2.1329 283 

2.4098 450 

2.7196 237 

3.0658 046 

3.4522 712 

14 

1.9799 316 

2.2609 040 

2.5785 342 

2.9371 936 

3.3417 270 

3.7974 983 

15 

2.0789 282 

2.3965 582 

2.7590 315 

3.1721 691 

3.6424 825 

4.1772 482 

16 

2.1828 746 

2.5403 517 

2.9521 638 

3.4259 426 

3.9703 059 

4.5949 730 

17 

2.2920 183 

2.6927 728 

3.1588 152 

3.7000 181 

4.3276 334 

5.0544 703 

18 

2.4066 192 

2.8543 392 

3.3799 323 

3.9960 195 

4.7171 204 

5.5599 173 

19 

2.5269 502 

3.0255 995 

3.6165 275 

4.3157 011 

5.1416 613 

6.1159 090 

20 

2.6532 977 

3.2071 355 

3.8696 845 

4.6609 571 

5.6044 108 

6.7275 000 

21 

2.7859 626 

3.3995 636 

4.1405 624 

5.0338 337 

6.1088 077 

7.4002 499 

22 

2.9252 607 

3.6035 374 

4.4304 017 

5.4365 404 

6.6586 004 

8.1402 749 

23 

3.0715 238 

3.8197 497 

4.7405 299 

5.8714 637 

7.2578 745 

8.9543 024 

24 

3.2250 999 

4.0489 346 

5.0723 670 

6.3411 807 

7.9110 832 

9.8497 327 

25 

3.3863 549 

4.2918 707 

5.4274 326 

6.8484 752 

8.6230 807 

10.8347 059 


Subtract $1 from the amount in this table to find the interest. 























COMPOUND INTEREST 333 

706. 

Table showing Amount of SI Deposited Annually at Com¬ 
pound Interest for any Number of Years not Exceeding 
Twenty-five. 


Peri¬ 

ods 

2 per cent 

3 per cent 

4 per cent 

per cent 

S per cent 

6 per cent 

1 . 

1.02 

1.03 

1.04 

1.045 

1.05 

1.06 

2. 

2.0604 

2.0909 

2.1216 

2.137025 

2.1525 

2.1836 

3. 

3.121608 

3.183627 

3.246464 

3.278191 

3.310125 

3.374616 

4. 

4.204040 

4.309136 

4.416323 

4.470710 

4.525631 

4.637093 

5. 

5.308121 

5.468410 

5.632975 

5.716892 

5.801913 

5.975319 

6. 

6.434283 

6.662462 

6.898294 

7.019152 

7.142008 

7.393838 

7. 

7.582969 

7.892336 

8.214226 

8.380014 

8.549109 

8.897468 

8. 

8.754628 

9.159106 

9.582795 

9.802114 

10.026564 

10.491316 

9. 

9.949721 

10.463879 

11.006107 

11.288209 

11.577893 

12.180795 

10. 

11.168715 

11.807796 

12.486351 

12.841179 

13.206787 

13.971643 

11. 

12.412090 

13.192030 

14.025805 

14.464032 

14.917127 

15.869941 

12. 

13.680332 

14.617790 

15.626838 

16,159913 

16.712983 

17.882138 

13. 

14.973938 

16.086324 

17.291911 

17.932109 

18.598632 

20.015066 

14. 

16.293417 

17.598914 

19.023588 

19.784054 

20.578564 

22.275970 

15. 

17.639285 

19.156881 

20.824531 

21.719337 

22.657492 

24.672528 

16. 

19.012071 

20.761588 

22.697512 

23.741707 

24.840366 

27.212880 

17. 

20.412312 

22.414435 

24.645413 

25.855084 

27.132385 

29.905653 

18. 

21.840559 

24.116868 

26.671229 

28.063562 

29.539004 

32.759992 

19. 

23.297370 

25.870374 

28.778079 

30.371423 

32.065954 

35.785591 

20. 

24.783317 

27.676486 

30.969202 

32.783137 

34.719252 

38.992727 

21. 

26.298984 

29.536780 

33.247970 

35.303378 

37.505214 

42.392290 

22. 

27.844963 

31.452884 

35.617889 

37.937030 

40.430475 

45.995828 

23. 

29.421862 

33.426470 

-38.082604 

40.689196 

43.501999 

49.815577 

24. 

31.030300 

35.459264 

40.645908 

43.565210 

46.727099 

53.864512 

25. | 

32.670906 

37.553042 

43.311745 

46.570645 

50.113454 

58.156383 


707. Compound interest is interest on the principal and its un¬ 
paid interest, the principal and interest being combined at regular 
intervals. The intervals may be a year, six months, or three 


months, etc. 

708. While compound interest is not generally collectible by 
law, its acceptance by a creditor is not usurious. 















334 


INTEREST 


709. Savings banks generally pay compound interest on de¬ 
posits. Premiums in life insurance are determined by the applica¬ 
tion of the principles of compound interest. Returns from an 
investment in bonds are reckoned by compound interest. 

710. To find compound interest. 

Find the compound interest on $1000 for 3 years at 6%. 

6% of $1000 = $60, first year’s interest. 

$1000 + $60 = $1060, new principal, second year. 

6% of $1060 = $63.60, second year’s interest 

$1060 + $63.60 = $1123.60, new principal, third year. 

6% of $1123.60 = $67.42, third year’s interest. 

$1123.60 + $67.42 = $1191.02, amount at end of third year. 
$1191.02 — $1000 = $191.02, compound interest. 

Or, by using the table: 

1000 X $1.191016 = $1191.016 = $1191.02, amount. 

$1191.02 — $1000 = $191.02, compound interest. 

Find the interest on $1000 for the first year, and add it to the $1000, making 
a new principal of $1060 for the second year. Find the interest on $1060 for 
the second year and add it to $1060, which gives $1123.60, the new principal for 
the third year, and so on. From the total amount due at the expiration of the 
three years deduct the original principal, $1000; the remainder is the compound 
interest. 

By using the compound interest table, the labor of computation is much less¬ 
ened. First find the amount of $1 for 3 yr. at 6%, by looking in the 6% column 
and at the number opposite the third year, as indicated in the margin. The 
number is $1.191016. Since $1 amounts to $1.191016, $1000 will amount to 
1000 times $1.191016, or $1191.02. Deducting the principal gives $191.02, the 
compound interest. 

PROBLEMS 

711. Find, without the use of the table, the compound interest 
on; 

1. $800 for 5 yr. at 5%. 

2. $900 for 6 yr. at 6%. 

3. $1200 for 4 yr. at 8%. 

4. $2000 for 3 yr. at 6%, compounded* semiannually. 

5. $4000 for 2 yr. at 8%, compounded quarterly. 

*Find the interest for 1 yr., and add to the principal. 


SINKING FUNDS 


335 


If $100 is deposited annually at 5% compound interest, how 
much will it amount to in 20 yr.? 

$1 amounts to $34.7193. 

100 X $34.7193 = $3471.93, amount. 

Use the table in the following: 

6. A boy 16 yr. of age has $1500 deposited in a savings bank 
for him. The savings bank pays 4% interest, compounded semi¬ 
annually. What amount will he have to his credit when he is 
21 yr. of age, no withdrawals nor any more deposits having been 
made? 

7. A young man 25 yr. of age has his life insured for $2000 by 
taking out a 20-yr. endowment policy for which he pays annually 
$96.30. If, at the expiration of the 20 yr., he receives the face 
value of the policy, find the gain to the insurance company if 
money is worth 4% compound interest to them. (See table, 
page 333.) 

8. If the young man in problem 7 had died at the age of 35, 
would the insurance company have gained or lost, and how much? 

9. A boy on his twelfth birthday started a savings bank account 
by depositing $25. If he deposits $25 every 6 mo. thereafter until 
he is 21 yr. of age, what amount will he have to his credit, the bank 
paying 4% interest compounded every 6 mo.? 

10. What sum deposited in a savings bank paying 4% compound 
interest will amount to $5000 in 25 yr.? 

SINKING FUNDS 

712. In financing large undertakings, as the building of railways 
or the erection of large buildings, etc., immediate funds are pro¬ 
vided by borrowing the money and issuing bonds therefor. The 
bonds are payable in 10, 15, 20, or more years, as the case may be. 
To meet these bonds at maturity, a sufficient sum of money is 
invested each year to amount, with compound interest, to the 
face value of the bonds. The sum thus set aside is called a sinking 
fund. 

Note. All problems in sinking funds are based on compound interest. 


The table, page 333, shows 
that $1 deposited each year for 
20 yr. at 5% amounts to 
$34.7193. $100 will amount 
to 100 times $34.7193, or 
$3471.93. 


336 


INTEREST 


713. Table showing Amount of Annuity* of $1 at End of 

Each Period 


Peri¬ 

ods 

3 per cent 

4 per cent 

41 per cent 

5 per cent 

6 per cent 

1 . 

1 . 

1 . 

1 . 

1 . 

1 . 

2. 

2.03 

2.04 

2.045 

2.05 

2.06 

3. 

3.0909 

3.1216 

3.137025 

3.1525 

3.1836 

4. 

4.183627 

4.246464 

4.278191 

4.310125 

4.374616 

5. 

5.309136 

5.416323 

5.470710 

5.525631 

5.637093 

6. 

6.468410 

6.632975 

6.716892 

6.801913 

6.975319 

7. 

7.662462 

7.898294 

8.019152 

8.142008 

8.393838 

8. 

8.892336 

9.214226 

9.380014 

9.549109 

9.897468 

9. 

10.159106 

10.582795 

10.802114 

11.026564 

11.491316 

10. 

11.463879 

12.006107 

12.288209 

12.577893 

13.180795 

11. 

12.807796 

13.486351 

13.841179 

14.206787 

14.971643 

12. 

14.192030 

15.025805 

15.464032 

15.917127 

16.869941 

13. 

15.617790 

16.626838 

17.159913 

17.712983 

18.882138 

14. 

17.086324 

18.291911 

18.932109 

19.598632 

21.015066 

15. 

18.598914 

20.023588 

20.784054 

21.578564 

23.275970 

16. 

20.156881 

21.824531 

22.719337 

23.657492 

25.672528 

17. 

21.761588 

23.697512 

24.741707 

25.840366 

28.212880 

18. 

23.414435 

25.645413 

26.855084 

28.132385 

30.905653 

19. 

25.116868 

27.671229 

29.063562 

30.539004 

33.759992 

20. 

26.870374 

29.778079 

31.371423 

33.065954 

36.785591 


714. To find an annuity which will amount to a given debt. 

1. The city of X put in a system of city waterworks at an expense 
of $150,000, and sold municipal bonds maturing in 10 yr. to pay for 
the works. What sum must the city set aside each year at 4% to 
redeem the bonds at maturity? 

By referring to the above table, an annuity of $1 for 10 yr. at 
4% is found to amount to $12.006107. If $1 amounts to 
$12.006107 in ten years, it will require as many dollars to amount 
to $150,000 as $12.006107 is contained times in $150,000, or 
$12,493.64. 

*A series of equal payments made at regular periods is called an annuity. If an annuity 
is invested, so that, with its interest accumulation it will be equal to a given amount or debt, 
is called a sinking fund. 














SINKING FUNDS 


337 


2. Jan. 1, 1924 a man borrows $1000 for 5 yr. at 4% and agrees 
to pay principal and interest (compounded annually) in five 
equal payments. What annual payment is required? 

1000 X $1.2166529 = $1216.6529, amount of $1000 in 5 yr. 
$1216.65-5-$5.416323 = $224,627, or $224.63. 

Reference to the compound interest table, page 332, shows the amount of SI 
at 4% for 5 yr. to be $1.2166529. $1000 amounts to 1000 times $1.2166529, or 
$1216.65. 

By the table on page 336, an annuity of $1 for 5 yr. at 4% amounts to 
$5.416323. Dividing the total amount due by $5.416323 gives the annual 
payment as $224,627, or $224.63. 

Proof of this solution is shown in the following: 


Schedule of Amortization 


Dates 

Annual 

Payment 

Interest 
on Balance 

Amortization 

Principal 

Unpaid 

Jan. 1, 1924 . . . 
Dec. 31, 1924 . . . 

$224.63 

$40.00 

$184.63 

$1000.00 

815.37 

Dec. 31, 1925 . . . 

224.63 

32.62 

192.01 

623.36 

Dec. 31, 1926 . . . 

224.63 

24.94 

199.69 

423.67 

Dec. 31, 1927 . . . 

224.63 

16.95 

207.68 

215.99 

Dec. 31, 1928 . . . 

224.63 

8.64 

215.99 

000.00 


$1123.15 - 

- $123.15 = 

$1000.00 



Each annual payment is composed of two items: first, the 
interest on the unpaid principal; and second, a payment on account 
of the principal. The column headed “Amortization” shows the 
amount of the several payments on account of the principal. The 
total amount paid equals the sum of the interest and the principal. 

Note. The apparent discrepancy between the total amount paid ($1123.15), 
and the amount of $1000 at 4% for 5 yr. ($1216.65), is due to the fact that 
$1216.65 is the amount that would have been due and payable if no payments 
had been made till Dec. 31, 1928. 

PROBLEMS 

715. l. For the erection of new school buildings, the city of M 
issued $75,000 municipal bonds bearing 4|% interest, and payable 
in 15 years. What amount of taxes should be levied each year to 

VAN TUYL’S NEW COMP. AR.— 22 
















338 


INTEREST 


provide a sinking fund which, at 4% compound interest, will be 
sufficient to redeem the bonds at maturity and to pay the interest ? 

2. City bonds to the amount of $10,000,000, maturing in 20 yr., 
were issued Oct. 1, 1924. . To pay the bonds at maturity a sink¬ 
ing fund earning 4|% is provided. What should be the annual in¬ 
vestment in the sinking fund? 

3. The government of Ontario, Canada, loans money to farmers 
in even hundreds of dollars to be used in draining their land. The 
loan and 4% compound interest is to be repaid in 20 equal annual 
installments. Find the annual payment per $100 required to 
cancel the loan. Prepare a schedule of amortization showing the 
amount paid on account of the interest and of the principal each 
year. 

4. The Southern Power Company has $3,000,000 of 5% first 
mortage bonds outstanding. If these bonds are dated Mar. 1, 
1924, and are payable on Mar. 1, 1944, what amount must be set 
aside each year at 4|% to meet them at maturity? 

5. A manufacturing company has installed a new piece of 
machinery at a cost of $6000. It is estimated that it will have to 
be replaced in 6 years. If the worn out machine can be sold as 
junk for $500, what sum must be set aside each year so that, with 
interest at 5%, a fund will be provided for purchasing a new 
machine? 

6. Set up a schedule of accumulation to show the amount of the 
fund (including interest) each year. 

7. A man bought a house and lot for $15,000, paying $5000 in 
cash. He agreed to pay the balance, with interest at 5%, com¬ 
pounded annually, in 5 equal annual installments. Find the value 
of each installment. 

8. Assume in problem No. 7 the agreement had been to pay the 
interest on the $10,000 annually and to pay the $10,000 in one sum 
at the end of the 5 years. Assume also that the purchaser of the 
house provides the $10,000 by means of a sinking fund which 
accumulates at 5% compound interest. Would it cost more or 
less by this plan than by the method of payment in No. 7, and 
how much? 


PARTIAL PAYMENTS 


339 


PARTIAL PAYMENTS 

716. When part of note, draft, or other obligation is paid, such 
payment is called a partial payment. 

717. Any partial payment of a note or draft should be indorsed 
on the note or draft, thus: 

UNITED STATES RULE 

718. The United States Rule (so 
called because it has been approved by 
the United States Supreme Court) is 
based on the following principles: 

1. Any payment must first be applied 
to the paying of accrued interest. If the 
payment exceeds the amount of interest 
due, the principal sum is reduced by the 
amount of the excess. 

2 . Interest must not be charged upon 
interest. 

719. This rule is generally used when partial payments are made 
on interest-bearing notes having more than one year to run. (Use 
compound time.) 

720 To find the balance due by the United States Rule. 

$3000.00 New York, N.Y., Sept. 8, 1923. 

Two years after date, I promise to pay to the order of A. C. Losey, Three 
thousand T °o 0 ^ Dollars, with interest at 6%. Value received. 

John Carpenter. 

The following payments were indorsed on the above note: Jan. 
20, 1924, $1200; Nov. 10, 1924, $75; and Apr. 4, 1925, $500. 
What was the balance due at maturity? 


Principal sum due .$3000.00 

Int. from Sept. 8, 1923, to Jan. 20, 1924 (4 mo. 12 da.) 66.00 

Amount due Jan. 20, 1924 . 3066.00 

Payment, Jan. 20, 1924 . 1200.00 

Balance due, to draw int. from Jan. 20, 1924 . . . 1866.00 


Int. from Jan. 20, 1924, to Nov. 10, 1924 (9 mo. 20 da.) 90.19 


Rec*d on this note 

Apr. 1 , 1924, $500.00 
July 10, 1924, $675.50 
Sept. 15, 1924, $853.75 










340 


INTEREST 


Since the payment is less than the accrued interest, 
the balance due remains unchanged. 

Int. on $1866 from Nov. 10, 1924, to Apr. 4, 1925 


(4 mo. 24 da.). 44.78 

Amount due Apr. 4, 1925 .$2000.97 

Payments, Nov. 10,1924, and Apr. 4, 1925 ($75 +$500) 575,00 

Balance due, to draw int. from Apr. 4, 1925 .... 1425.97 
Int. from Apr. 4, 1925, to Sept. 8, 1925 (5 mo. 4 da.) . 36.60 

Balance* due Sept. 8, 1925, date of maturity .... $1462.57 


The principal sum, $3000, draws interest from the date of the note, Sept. 8, 

1923, to the date of the first payment, Jan. 20, 1924, a period of 4 mo. 12 
da. The interest is $66, making a total amount due of $3066. Deducting 
the payment of $1200, leaves a balance of $1866 to draw interest from Jan. 20, 

1924. From Jan. 20, 1924, to the date of the next payment, Nov. 10, 1924, is 
9 mo. 20 da., and the interest accrued in that time is $90.19. But the payment 
made on Nov. 10, 1924, is only $75, not large enough to cancel the interest; 
hence the balance due, $1866, remains unchanged, and continues to draw 
interest to the date of the next payment, Apr. 4, 1925, or 4 mo. 24 da. The 
interest on $1866 for 4 mo. 24 da. is $44.78, making a total amount due of 
$2000.97. Deducting the sum of the two payments, $75 and $500, leaves 
$1425.97, the balance to draw interest till the date of maturity, Sept. 8, 1925. 
From Apr. 4,1925, to Sept. 8, 1925, is 5 mo. 4 da., and the interest on $1425.97 
for 5 mo. 4 da. is $36.60, which makes the amount due at maturity equal to 
$1462.57. 

PROBLEMS 

721, Find the balance due at maturity on each of the following 
notes, payments having been indorsed as indicated: 



Date 

Time to Run 

Face 

Int. 

Indorsements 

1. 

May 1,1924 

3 yr. 

$2500 

6% 

Dec. 1, 1924, $ 500 
June 21,1925, 300 

Jan. 15,1926, 1000 

2. 

Aug. 7,1923 

4 yr. 

$5000 

5% 

Sept. 1,1924, $1200 
Aug. 15,1925, 1500 
Dec. 20,1926, 1000 

S. 

Jan. 10, 1924 

2 yr. 6 mo. 

$4000 

7% 

Apr. 20,1924, $ 500 
July 30,1924, 50 

Dec. 10,1925, 1000 

















PARTJAL PAYMENTS 


341 


722. The Merchants’ Rule is frequently used to find the balance 
due when partial payments are made on interest-bearing notes in 
which the time is less than one year. It is based on the following 
principles: 

1. The face of the note draws interest from its date to the time of 
settlement. 

2. Each payment draws interest from its date to the time of settle¬ 
ment. 

723. To find the balance due by the Merchants’ Rule. 

$1800.00 New York, N.Y., Oct. 8, 1924. 

Ten months after date I promise to pay to the order of J. K. Ramsey, 
Eighteen hundred T °A Dollars, with interest at 6%. Value received. 

Theodore A. Bidwell. 

On the above note the following payments were indorsed: Dec. 
20,1924, $300; Feb. 1, 1925, $600; June 1, 1925, $400. Find the 
balance due at maturity. 


Compound Time 


Date 

1924 

Face 

Time 

Int. | 

Dates 

1 1924 

Payments 

Time 

Int. 

Oct. 8. 

$1800 

10 mo. 

$90 

Dec. 20 

1925 

$300 

7 mo. 18 da. 

$11.40 





Feb. 1 

600 

6 mo. 7 da. 

18.70 





June 1 

400 

2 mo. 7 da. 

4.47 


$1800 


$90 


$1300 


$34.57 


$1800 + $90 = $1890; $1300 + $34.57 = $1334.57. 
$1890 — $1334.57 = $555.43, balance due at maturity. 


Exact Time 


Date 

1924 

Face 

Time 

Int. 

Dates 

1924 

Pay¬ 

ments 

Time 

Int. 

Oct. 8 

$1800 

304 da. 

$91.20 

Dec. 20 

1925 


$300 

231 da. 

$11.55 





Feb. 1 


600 

188 da. 

18.80 





June 1 


400 

68 da. 

4.53 


$1800 


$ 91.20 



$ 1300 


$34.88 


$1800 + $91.20 = $1891.20; $1300 + $34.88 - $1334.88. 
$1891.20 - $1334.88 = $556.32. 








































342 


INTEREST 


Compound Time. The solution of problems in Partial Payments by the 
Merchants’ Rule is a direct application of the principles involved in Cash 
Balance. (See page 419.) The face of the note constitutes the debit side of 
the account and the several payments the credit side. The due date is Aug. 
8, 1925. The time of the note is 10 mo., and the interest amounts to 890, 
making a total amount due of $1890. From the time of the first payment, 
Dec. 20, 1924, to the due date, Aug. 8, 1925, is 7 mo. 18 da., and the interest 
on the payment, 8300, for that time is 811.40. In the same way the time and 
interest of the second payment is found to be 6 mo. 7 da. and 818.70, and for 
the third payment, 2 mo. 7 da. and 84.47, respectively. The total credit is 
the sum of the payments, 81300, plus the sum of the interest items, 834.57, 
or 81334.57. The balance due, Aug. 8, 1925, is the difference between 81890 
and 81334.57, or 8555.43. 

Exact Time . The solution by exact time is the same as by compound time 
except that the several interest periods are found by exact time. This makes a 
slight difference in the interest items, as shown in the solutions. 

Note. There is no law making the Merchants’ Rule a legal method. 
Hence, when it is to be used in practice, it should be agreed upon by the parties 
interested. 

Some business men use exact time, while others use compound time. The 
answers in this text are given for both methods. 

PROBLEMS 


724. Find the balance due at maturity on each of the following 
notes, payments having been indorsed as indicated: 



Date 

Time to Run 

Face 

I NT. 

Indorsements 

1. 

July 1,1924 

11 months 

81000 

6% 

Sept. 20,1924,8200 
Dec. 18,1924, 300 
Apr. 1,1925, 300 

2. 

Oct. 8,1924 

9 months 

1500 

5% 

Dec. 15,1924, 200 
Jan. 20,1925, 400 
June 14,1925, 500 

3. 

Mar. 1, 1924 

i 

10 months 

1360 

6% 

May 12,1924, 75 

June 15,1924, 90 

Aug. 20,1924, 120 
Oct. 11,1924, 250 
Nov. 4,1924, 300 

4. 

Apr. 30,1924 

8 months 

930 

7% 

June 10,1924, 80 

July 13,1924, 125 
Sept. 15,1924, 250 
Nov. 20,1924, 200 














TAXES 


725. Who pays the salary of the governor, the legislators, the 
mayor, the policemen, and the teachers, in your state and city, or 
town? 

How is the money raised for such purposes? 

Ask the assessors of your town to explain to you the method of 
raising money for the above-named purposes. 

726. Money charged against persons and their property for the 
payment of public expenses is called a tax. 

727. Taxes are of two kinds—direct and indirect. 

728. A direct tax is a tax levied on a person, his property, or his 
business. 

729. A tax on a person is called a poll tax. 

730. A tax on property, real or personal, is called a property tax. 

731. A tax on a person’s business is called a license fee. 

732. An indirect tax is a tax (called a duty) on imported goods 
(see U. S. Customs, page 406) or a tax (called excise or internal 
revenue) on the manufacture of tobacco products. The tax 
need not be paid on tobacco that is exported. 


HOW TAXES ARE APPORTIONED 


733. Suppose that in the state of X there are 4 counties, whose 
property values are as follows: 


County A 
County B 
County C 
County D 
Total 


$15,000,000 

25,000,000 

27,000,000 

33,000,000 

$100,000,000 


Suppose, also, that County A is divided into 3 towns, whose 


values are: 
Town E 
Town F 
Town G 
Total 


$5,000,000 

4,000,000 

6,000,000 

$15,000,000 


The amount to be raised in the state for state expenses, exclusive 


343 













344 


TAXES 


of the revenue received from licenses, permits, etc., is, say, 
$100,000. This amount is apportioned among the several counties 
in the ratio of their valuation. The valuation of County A is 
15% of that of the entire state; hence, County A must raise 15% 
of $100,000, or $15,000, for its share of the state tax. 

The county tax is $45,000, to which the state tax of $15,000 is 
added, making a total of $60,000 to be raised in County A for state 
and county expenses. This $60,000 is apportioned among the 
towns in the ratio of their valuation. The value of Town E is f 
that of County A; hence, Town E is charged with f of $60,000, or 
$20,000, its share of state and county taxes. To this amount is 
added the town tax of $70,000, which makes $90,000 to be raised 
in the town of E. To find the rate of tax, $90,000 is divided by the 
valuation of the town, $5,000,000, which gives .018 = If %. Each 
person owning taxable property in the town is then charged with 
a tax equal to If % of the valuation placed on his property. 

734. The persons appointed to make an estimate of the value of 
each person’s property, and to apportion the taxes in proportion 
to the value of each person’s property, are called assessors. 

735. The assessors make what is called an assessment roll. It is 
a list showing the names of property owners, a brief description of 
the property owned, its assessed valuation, and the tax thereon. 

736. The person who collects the tax from the property owners 
is called the collector. 

737. Solutions of problems in taxes depend upon the principles 
of percentage. 

The assessed valuation is the base. 

The tax rate is the rate. 

The tax is the 'percentage. 

738. The tax rate is frequently expressed as $1.80 per $100, 
instead of 1.8%. 

EXERCISES 

739. 1 . A tax of $.003 on a dollar is how how much on $100? 
on $1000? 

2 . What is the tax on $4500 at the rate of $.65 per $100? 

3. At $8.50 per $1000, what is the tax on $7800? 

4. Find the tax on $15,400 at 1.365%. 


TAXES 


345 


Find the tax on the following: 


5. $13,500 at .0065. 

6. $34,500 at .01467. 

7. $44,800 at .00895. 

8. $18,500 at .0102. 

9. $125,000 at .00882. 

10. $135,500 at .004. 

11. $500,000 at .0115. 


12. $240,000 at $1.64 per $100. 

13. $15,900 at $1,845 per $100. 

14. $32,600 at $13.26 per $1000. 

15. $16,400 at $11.15 per $1000. 

16. $69,500 at $.9875 per $100. 

17. $51,600 at $1,065 per $100. 

18. $12,600 at $12,853 per $1000. 


Oftentimes the collector is allowed a commission on all taxes 


collected, instead of a salary. 


740. 1 . Find the total tax paid by a man whose property is as¬ 
sessed for $13,800, at 1.25%, the collector’s commission being 1%. 
1.25% of $13,800 = $172.50, tax. 

1% of $172.50 = 1.73, collector’s commission. 

Total tax, $174.23. 


The tax is 11% of $13,800, or $172.50. Thecollector’s commission of 1 % is 
reckoned on the amount of tax, and then added to the tax, and charged against 
the property owner. 

2. The town of Wellsville has an asssessed valuation of $2,118,000- 
The total amount of tax to be collected is $18,917. There are 910 
polls at $1 each, and $2200 highway tax not to be charged on the 
valuation. Find the rate of taxation and the amount of tax paid 
by a farmer whose porperty is assessed for $6400, who pays for 1 
poll, and $3 highway tax. 

$910 + $2200 * = $3110, to be deducted from total tax. 

$18,917 — $3110 = $15,807, tax levied on property. 

$15,807 $2,118,000 = $.007463 = 7.463 mills on $1. 

$6400 X .007463 = $47.76, property tax. 

$47.76 + $1 + $3 = $51.76, total tax. 

To find the amount of the tax to be levied on the property of the town, de¬ 
duct the sum of the poll tax and the highway tax, which leaves $15,807. The 
tax on the property divided by the value of the property, gives the rate, which 
is 7.463 mills to the dollar. The property tax on a farm valued at $6400 is 
found by multiplying $6400 by .007463, which gives $47.76, to which is added 
$1, poll tax, and $3, highway tax, making a total tax of $51.76, 



346 


TAXES 


741 . l. The assessed valuation of the real and personal property 
in a certain town is $1,040,000. The total amount of tax to be col¬ 
lected is $7136. There are 206 polls at $1 each, and $1600, high¬ 
way tax. Find the rate of taxation, and the amount of tax paid by 
a man whose property is assessed at $10 500 and who pays for 1 
poll and a highway tax of $8. 

5 . The city of Albany, N. Y., appropriated $320,684 for the sup¬ 
port of the public schools. If the taxable property of the city is 
assessed at $74,718,000, what was the rate of the school tax? 

3. The tax rate in Albany for the same year was $1.94 per $100. 
What was the amount raised by tax ? The school tax was what per 
cent of the entire tax? 

In Albany, a discount of 1% is allowed on all taxes paid before 
the 10th of February; if paid on or after the tenth day of Febru¬ 
ary and before the first day of March, |% discount is allowed; if 
paid on or after Mar. 1 and before Apr. 1, no discount is allowed; 
if paid on or after Apr. 1, \% is added on the first day of each 
month for the remainder of the year. Under these conditions find 
the amount of tax paid on the following: 

4 . House assessed at $2900, tax paid Feb. 8. 

5 . House assessed at $6300, tax paid Feb. 10 . 

6 . Store assessed at $15,000, tax paid Mar. 31 . 

7 . Factory assessed at $45,500, tax paid July 30. 

8. Store assessed at $38,700, tax paid D^c. 20. 

9 . The assessed valuation of real estate, and the tax rate in 
the several boroughs of New York City for 1923 were as follows: 


Borough 

Valuation 

Tax Rate 

Amount op Tax 

Manhattan. 

$6,177,890,668 

.0274 


The Bronx. 

926,682,418 

.0274 


Brooklyn. 

2,536,590,061 

.0274 


Queens. 

804,004,439 

.0274 


Richmond. 

150,897,987 

.0276 



Find the amount of tax for each, borough. 













TAXES 


347 


10. Find the per cent of the total valuation in each borough. 

11. What per cent of the tax is levied in each borough? 

In the city of New York one half of the tax on real estate and all 
the tax on personal property are due and payable on and after May 
1. The other half of the real estate tax is due and payable on and 
after Nov. 1. A discount of 4% per annum is allowed on the second 
half of the real estate tax if paid before Nov. 1, provided the first 
half has been paid. Interest at 7% per annum from May 1 is 
added to all payments of the first half of the real estate tax and all 
personal taxes paid on and after June 1. Interest at 7% per 
annum from Nov. 1 is added to all payments of the second half of 
real estate tax on and after Dec. 1. 

Find the amount of tax due and payable in the city of New York 
on the following properties (discounts and penalties as stated 
above): 


Property 

Assessed 

Borough 

Rate 

Date of 

Payment 


Valuation 



First Half 

Second Half 

12 . Metropolitan 

Life Bld’g 

13. N. Y. C. & 

H. R. R. 

14. Long Island 

Railroad Co. 

15 . Apartment 

house 

16 . House and lot 

$12,415,000 

17,605,400 

16,756,500 

75,000 

12,500 

Manhattan 

Bronx 

Queens 

Brooklyn 

Richmond 

See Problem $ 9 
page 346 

June 18, ’23 

July 1, ’23 

May 31, ’23 

Aug. 15, ’23 
Sept. 10, ’23 

Oct. 10/23 

Dec. 30, ’23 

Nov. 18, '23 

Mar. 21, ’24 
Sept. 10, ’23 


In St. Louis, Mo., the tax rate for 1920 was, for state purposes, 
18^; for schools, 78j£; and for city purposes, SI.59 per $100. 
Taxes are due Sept. 1. Taxes become delinquent after Dec. 31. 
The penalty on all delinquent taxes is 1% a month from date of 
delinquency till paid, provided they are paid before the Saturday 
preceding the first Monday in March. After said date all costs of 
collection are added. 

Under the above conditions find the total tax for the year 1920 












348 


TAXES 


on each of the following properties, if taxes were paid on the date 
given: 



Description of Property 

Valuation 

Date Tax was Paid 

17. 

4-story apartment house . 

$25,000 

Sept. 25,1920 

18. 

Lot 75' X 100'. 

5,500 

Feb. 18,1921 

19. 

12-story office building 

1,200,000 

Sept. 29,1920 

20. 

2-story frame dwelling 

5,000 

Dec. 30,1920 


In Denver, Colo., taxes are paid in two payments of 50% each. 
Taxes are due Jan. 1 of each year. The first half is delinquent and 
draws interest at 1% a month on and after Mar. 1. The second 
half is delinquent and draws interest at 1|% a month on and after 
Aug. 1. 

If the state tax is 21^, and the city tax is $1.38 per $100, find 
the total tax paid on the following: 

21. Store valued at $45,000, both installments paid July 20. 

22. Dwelling valued at $12,500, first installment paid Feb. 27; 
second installment paid Dec. 1. 

23. Hotel valued at $500,000, first installment paid July 20; 
second installment paid Oct. 10. 

On the following page is a form of tax bill used in San Francisco, 
Cal. Note that all the personal property tax is due and payable 
at the same time the first half of the real estate tax is due. Study 
this form carefully and verify the results given. 

Using the information given in the tax bill on the next page, 
find the tax paid in San Francisco on the following properties: 



Real Estate 
Valuation 

Personal 

Property 

Valuation 

First Installment 
Paid 

Second Installment 
Paid 

24. 

$1,960 


Nov. 20,1920 

April 10,1921 

25. 

$22,500 

$5500 

April 12,1921 

April 12,1921 

26. 

$13,600 

$7700 

Dec. 15,1920 

May 18,1921 

27. 

$5,500 

$2300 

June 1,1921 

June 1 , 1921 

28. 

$3,200 


Dec. 1 , 1920 

April 25,1921 
























TAXES 


349 


Mali ill Clicks ar,i Uogiy Ordm payabli to 

EDWARD F. BRYANT 

City and County Tax Collector 
City hall, San Francisco, Calif. 



Ift Installment due Oct. It, 1920. 

15 9& added Dec. 6, 1920, at 6 r. m. 
5 > added April 2S, 1921* at 6 r. m. 


REAL ESTATE AND SECURED PERSONAL 
PROPERTY TAXES 
TAX COLLECTOR NOT 
RESPONSIBLE FOR ERRORS 

EXAMINE THIS DILL. CAREFULLY 

Tax Payers Please Take Notice 

Checks, Money Orders, Drafts, etc, will be 
accepted, provided your remittance Is payable 
In San Francisco and received In this ofllce prior 
to the date set below lor each collection: 

1st Installment, November 29th, 1920 
2nd “ April 18th, 1921 

Postage stamps will not be accepted In full 
or part payment of taxes. 


Ofllce Open Evenings 
Until 9 P. M. 
During Last Week 
of Each Collection 



2nd Installment due Jan. 3, 1921. 

S* added April 2$, 1921, at 6 P. M. 
and 50c. additional for costs. 


NO RECEIPT VALID UNLESS PAID AT THE OFFICE, OFFICIALLY STAMPED, DATED, RECEIPTED AND CREDIT TAG OR TAGS TORN OFF 
Tax payers have the optioo of paying both Installments when the first installment falls due. 

I ol all leal Estate Is descried by Lot and Block oamber according to Assessment Map adopted by the Board ol Saperrtsors, November 29, 1911. 

Copies of tbe Mop are eo flk In the office ol Ibe Tax Collector, Assessor. Recorder and Aodltor. 

Compart Lot and Block Numbers With Last Years BUL EDWARD F. BRYANT. TaX Collector 


Real Estate Assessed to 
Personal Property Assessed to 



y 




At 




Street 


VOL. 

Z 

PAOK 

LOT 

BLOCK 

TAX BILL 
NO. 

VALUE or 
LAND 

VALUE or 
IMPROVEMENT 

PERSONAL 
PROPERTY VALU¬ 
ATION 

PERSONAL 
PROPERTY TAX 
ENTIRE 



T 

#7 


rffJo 


* 

» 







/] 


_ rl 


HI 

_ 











/i 


EA 

;a 

1 • T 

( N 

ITS 

FIRST INSTALLMENT 

or riRST K.or tax 

SECOND INSTALLMENT 
OR LAST Vi OP TAX 










-L 

On 

r 

o 


$ 


* 



& 

lountv 

ax Rat* 83 

18 


- 


i— 

□ W 


/ f 0 

L 0 

</? 0 



lor Its 

:al vear 

1920-19Z 





% C f f 

Jr 

_ 



This bill Is all that Is required for both installments. Bring it with you 
then paying in April, as no other bill will be furnished. 


S* 


Costs 


Bring this bill when paying 2nd Installment. Do nol detach these tags, they belong lo the Tax Collector. Not credited on roll without these lags. 


Assessed 1 


VOL. 

P*0i 

^ LOT 

BLOCK 


</z 

'A 

/VI 








TAX SILLNO. 


nr 

(TT 



is 

~7Z 



^7 U= 






5)0 


Costs 


IQ. 



The rates in the table on the following page apply to the $100 
assessed valuation, which is one third of the full value. 

Taxes for 1921 are collectible in 1922. 

On general taxes a penalty of 1% is charged after May 1, 2% 
after June 1, 3% after July 1, and so on, until paid. In addition 





















































































































350 


TAXES 


to this, advertising and copying costs of 19^ for each lot or part 
of a lot, and 29 i for each tract of land, are added to unpaid general 
taxes and special assessments, early in May. 

The following table shows how the tax rate is apportioned in 
Chicago, Ill.: 

Tax Rates for 1921 


Town 

Total 

Rate 

State 

County 

Forest 

Pre¬ 

serve 

Sani¬ 

tary 

City 

School 

Town 

Park 

Lincoln 

Park 

Bond 

West . 

5.57 

.40 

.52 

.06 

.17 

2.29 

1.62 


.51 


South. 

5.39 

.40 

.52 

.06 

.17 

2.29 

1.62 


.33 


North 

5.89 

.40 

.52 

.06 

.17 

2.29 

1.62 

.06 

.71 

.06 

Hyde Park 

5.39 

.40 

.52 

.06 

.17 

2.29 

1.62 


.33 


Lake . 

5.39 

.40 

.52 

.06 

.17 

2.29 

1.62 


.33 


Lake View 

5.88 

.40 

.52 

.06 

.17 

2.29 

1.62 

.03 

.73 

.06 

Rogers Pk. 

5.06 

.40 

.52 

.06 

.17 

2.29 

1.62 




Jefferson . 

5.06 

.40 

.52 

.06 

.17 

2.29 

1.62 





Using the rates and information given above, find the total tax 
on the following properties: 



Description of Property 

Assessed 

Valuation 

t 

Town 

Date of Payment 

29. 

Office building 

$150,000 

North Town 

April 20,1922 

30. 

3-story brown stone dwell¬ 
ing . 

90,000 

Hyde Park 

May 20, 1922 

31. 

Vacant lot 100' X 100'. 

15,000 

Jefferson 

July 30,1922 

32. 

A tract of land containing 
17 acres .... 

17,000 

Lake 

June 25,1922 






























INCOME TAX 


351 


INCOME TAX 

742. In order that the burden of taxation for national expenses 
may be more equitably distributed among the people, the Federal 
Government levies a graduated tax on the income of individuals 
who are citizens or residents of the United States. 

For the purpose of the normal tax certain exemptions are al¬ 
lowed. A single person is allowed SI000; and the head of a family 
or a married person living with husband or wife is allowed S2500, 
unless the net income is in excess of $5000, in which case the ex¬ 
emption is $2000. A husband and wife living together shall re¬ 
ceive only one exemption, but if they make separate returns, the 
exemption may be taken by either of them or it may be divided 
between them. A further exemption of $400 is allowed for each 
dependent child under 18 years of age receiving his support from 
the taxpayer. 

The normal tax is 4% of the first $4000 in excess of allowable 
deductions, and 8% of the excess over $4000. 

In addition to the normal tax, a surtax is imposed upon the net 
income of every individual taxpayer in excess of $6000. 

A partial list of rates of surtax follows: 

1% on net income from $6000 to $10,000 
2% on net income from 10,000 to 12,000 
3% on net income from 12,000 to 14,000 
4% on net income from 14,000 to 16,000 
and so on, adding 1%* to the rate for each additional $2000 of 
net income until the net income is $100,000, when the rates are: 
48% on net income from $100,000 to $150,000 
49% on net income from 150,000 to 200,000 
50% on net income over 200,000 

Example. 

A man's gross income for 1 year amounts to $18,750.75. His 
general expenses and other allowable deductions amount to 
$6475.40. Find the amount of the tax he should pay if he is 
married and has 2 children under 18 years of age. 

♦Except that there is no 7% or 14% rate. 


352 


TAXES 


His total income is. 

The allowable deductions are . . 

His net income is. 

His total exemption is . . . . 

His taxable income is. 


$18,750.75 

6,475.40 

$12,275.35 

2,800.00 

. . . $9,475.35 


The normal tax is 


4% of $4000 . . . 

, . . $160.00 


8% of $5475.35 . . 

. . 438.03 

$598.03 

The surtax is 

1% of $4000 . . . 

, . . $40.00 


2% of $2000 . . . 

. . 40.00 


3% of $275.35 . . . 

. . 8.26 

88.26 

The total tax is ... . 




PROBLEMS ' 

743. 1 . Mr. Ballard is a single man and has a salary of $6400. 
His allowable deductions for interest and benevolences amount 
to $495. How much income tax should he pay? 

2. A married man, having three young children, has a salary 
and other income of $9864.60. He is allowed to make deductions 
as follows: Taxes, $248.72; interest on indebtedness, $375; losses 
in business, $928.75; and benevolences, $532.50. Determine the 
amount of his income tax. 

3. A husband and wife, having two children under 18 years of 
age, are each in business and have separate incomes. The hus¬ 
band’s net income is $7490.50 and the wife’s is $4743.27. Find 
the amount of tax paid: 

(а) If the incomes are combined and only one return is made; 

(б) If husband and wife make separate returns, and each claims 
exemption for one child; 

(c) If husband and wife make separate returns, and the hus¬ 
band claims exemption for both children. 

Note: A State Income Tax is levied in some states. The laws differ 
in the several states. Classes interested should obtain information from their 
state capital covering local conditions. 













INSURANCE 

744. Insurance is an agreement by one party, for a consideration, 
to indemnify another party for losses arising from certain stipu¬ 
lated causes. 

The insurance company, or its authorized agent, is the party 
agreeing to pay in case of loss. 

745. The agreement or contract between the parties is called the 

policy. 

746. The consideration paid by the insured is called the pre¬ 
mium. It may be a certain per cent of the face of the policy, or it 
may be a specified amount per $100. 

747. The term of insurance may be any length of time, but is 
generally one or more years. 

748. OT the many kinds of insurance, only fire insurance and life 
insurance will be discussed in this text. 

FIRE INSURANCE 

749. Fire insurance treats of agreements to indemnify the in¬ 
sured for loss by fire. 

750. “Loss by fire” is held to cover not only loss of property 
actually burned, but also loss or damage resulting from the use 
of chemicals or water used in extinguishing the fire, and from 
smoke. Fire caused by lightning comes within the meaning of 
the term “loss by fire.” 

751. The fire insurance policies of all companies in Connecticut, 
New Jersey, New York, and Pennsylvania, are uniform, and 
contain the “New York Standard (80%) Average Clause,” which 
reads in part as follows: “This Company shall not be liable 
for a greater proportion of any loss or damage to the property 
described herein than the sum hereby insured bears to eighty 
per centum (80%) of the actual cash value of said property at 
the time such loss shall happen.” These policies also contain a 

VAN TUYL’S NEW COMP. AR.—23 


353 


354 


INSURANCE 


“Waiver Clause,” which reads: “In case of loss if the value of the 
property described herein does not exceed $2500, the 80% average 
or co-insurance clause shall be waived.” 

In other states there is the ordinary policy, which stipulates 
the amount of loss for which the insurance company is liable. 
Under the ordinary policy the company pays any loss not exceeding 
the face of the policy. 

752. In some states the policy contains an average clause 
which states that only such a part of the loss will be paid as the 
face of the policy bears to the value of the property insured. 

In all the states the fire insurance policies are quite similar. 

753. If more than one company insures the same property, each 
company pays only its pro rata share of any loss. 

754. Two general kinds of fire insurance policies are the valued 
policy and the open policy. 

755. In the valued policy, the value is stipulated and the com¬ 
pany agrees to pay on that specified amount. 

756. The open policy is used to cover goods in storage and else¬ 
where, the amount of the policy varying as the quantity of goods 
is increased or diminished by additions or withdrawals. The addi¬ 
tion and withdrawal of goods are recorded in an “Open Policy 
Book” retained by the company. Upon each receipt of goods, 
their nature and value are recorded in the policy book, and the 
premiun charged is based on the annual rate. 

In case of withdrawal in less than 1 yr. the comapny returns 
the unearned premium. 

757. A policy may be canceled at any time, either by the in¬ 
surer or by the insured, the company giving the insured 5 days* 
notice. If the insurance company cancels the policy, it will return 
to the insured such a part of the premium as the unexpired time of 
the policy is of the entire term of the policy. If, however, the in¬ 
sured cancels the policy, the company will return to him only the 
amount by which the premium paid is in excess of the premium 
reckoned at the short rate. The short rate is a correspondingly 
higher rate for a short period of time. 


FIRE INSURANCE 


355 


758. Insurance companies frequently issue a policy for 3 yr. 
for a premium equal to two and one half times the annual premium, 
and for 5 yr. for four annual premiums. 

759. The principles of percentage apply in insurance. 

The value of the policy is the base. 

The rate of premium is the rate. 

The premium is the percentage. 

760. To find the premium. 

If the property is insured for $18,000 at 20^ per $100 per annum, 
what is the annual premium? 

$18,000 = 180 hundreds of dollars. 

180 X $.20 = $36, the annual premium. 

Since the premium rate is on $100, first find the number of hundreds of dol¬ 
lars by pointing off 2 places in the $18,000, which gives 180 hundreds dollars. 
Tf the premium on $100 is $.20, on 180 hundreds it is 180 times $.20, or $36. 

PROBLEMS 


761. Find the premium on each of the following policies: 



Face op 
Policy 

Rate of 
Insurance 


Face op 
Policy 

Rate of Insurance 

1 . 

$13,200 

22 <k per $100 

8. 

24,600 

U%, less 10% 

2. 

9,600 

37^0 per $100 

9. 

55,000 

1%, less 10% and 5% 

3. 

16,400 

45^ per $100 

10. 

48,000 

55^ per $100, less 10% 

4. 

27,500 

65^ per $100 

11. 

20,000 

1% 

5. 

8,500 

\% 

12. 

24,500 

27^ per $100 

6. 

11,500 

f %, less 10% 

13. 

38,000 

£%, less 20% 

7. 

17,800 

f%, less 15% 

14. 

75,000 

44^ per $100, less 10% 


762. To find the amount paid by the insurer. 

l. Property valued at $25,000 is insured for $15,000 at f% per 
annum. Fire and water cause a loss of $12,000. What amount 
would be paid by the insurance company (a) under an ordinary 
policy? ( b ) under a co-insurance clause policy? (c) under the New 
York Standard (80%) Average Clause Policy? 

(a) $12,000, amount paid under an ordinary policy. 

15 000 _ 3 
\ 0 ) 2 5 0 0 (7 — 5 * 

f of $12,000 = $7200, amount paid under a co-insurance 
policy. 














356 


INSURANCE 


(c) 80% of $25,000 = $20,000. 

1 5000 _ 3 
20000 — 4 * 

f of $12,000 = $9000, amount paid under New York policy 

(а) Under an ordinary policy the company pays the full amount of loss up to 
the amount of the policy; hence, $12,000 would be paid in this case. 

(б) Under a co-insurance, or average clause policy, the part of the loss paid 
is the ratio of the face of the policy, $15,000, to the value of the property 
$25,000, or Minnb which is equal to Hence in this case the company would 
pay f of $12,000, or $7200. 

(c) Under the 80% average clause policy, the company pays such a part of 
the loss as the ratio of the face of the policy, $15,000, is to 80% of the value of 
the property, 80% of $25,000 = $20,000. The ratio of $15,000 to $20,000 is 
HtHHI = l- Hence the company would pay f of $12,000, or $9000. 

2. A stock of merchandise is insured in Company A for $5000, 
in Company B for $7000, and in Company C for $8000. In case of 
damage amounting to $10,000, how much will each company pay? 

$5000 + $7000 + $8000 = $20,000, total amount of insurance. 
= i; i of $10,000 = $2500, paid by Company A. 

-srcnnro- = 2 7 o, 2ir of $10,000 = $3500, paid by Company B. 

2 8 o 0 6 0 6 ° o = f;f of $10,000 = $4000, paid by Company C. 

The total insurance is $20,000. Each company pays such part of the loss as 
its risk is of the total risk. Hence, Company A pays \ of the loss, or $2500; 
Company B pays ^ of the loss, or $3500; and Company C pays f of the loss, 
or $4000. 

PROBLEMS 

763. 1. A dwelling valued at $9600 was insured for f of its value 
at f % and contents valued at $3200, for § of its value at £%. Fire 
causes a total loss of the building and a loss of $1400 on the con¬ 
tents. Under an ordinary policy, how much will the insurance 
company pay? 

2. Property insured under an ordinary policy for $15,000 at 30^ 
per $100 per annum is damaged by fire to the amount of $8000. 
What is the insurance company’s net loss if they have held the 
insurance 15 yr., compound interest 5%. 

3. A store and its contents were insured in Company X for 
$22,500 at 60^ per $100; in Company Y for $30,000 at f%; and in 
Company Z for $40,000 at 65^ per $100. The property was 





FIRE INSURANCE 


357 


damaged by fire and water to the amount of $40,000. What was 
each company’s net loss, they having held the insurance 7 yr., if 
money is worth 4J%, compounded annually? 

4. A man has his furniture insured for § of its value, in a policy 
containing an average clause. In case of a loss of $2400, how much 
should the company pay? 

5. A house valued at $15,000, and contents valued at $7500, are 
insured for § of their value. Fire causes a loss of $4000 on the 
house and $1500 on the contents. Find the amount payable under 
an 80% average policy. 

764 . The following table shows the short rates adopted by the 
“Western Union”: 


Standard Short Rate Scale for computing Premiums for Terms 
less than One Year, as adopted by the “Western Union” 

RULE. Take the percentage indicated in scale opposite the number of days risk is 
to run, on the premium for one year at given rate, and the result will be the premium 
earned in case of cancellation, or to be charged in case of short risks. 

1 Day . 

. . . 2 per cent annual prem. 

55 Days. 

29 per ct. an’l prem. 

2 Days . 

. . . 4 per cent annual prem. 

60 Days. 

30 per ct. an’1 prem. 

3 Days . 

. . . 5 per cent annual prem. 

65 Days. 

33 per ct. an’l prem. 

4 Days . 

. . . 6 per cent annual prem. 

70 Days. 

36 per ct. an ’1 prem. 

5 Days . 

... 7 per cent annual prem. 

75 Days. 

37 per ct. an’1 prem. 

6 Days . 

. . . 8 per cent annual prem. 

80 Days. 

38 per ct. an’l prem. 

7 Days . 

. . . 9 per cent annual prem. 

85 Days. 

39 per ct. an '1 prem. 

8 Days . 

. . . 9 per cent annual prem. 

90 Days or 3 mo. 

40 per ct. an '1 prem. 

9 Days . 

. . . 10 per cent annual prem. 

105 Days. 

45 per ct. an’l prem. 

10 Days . 

. . . 10 per cent annual prem. 

120 Days or 4 mo. 

50 per ct. an’1 prem. 

11 Days . 

. . . 11 per cent annual prem. 

135 Days. 

55 per ct. an’l prem. 

12 Days . 

. . . 12 per cent annual prem. 

150 Days or 5 mo. 

60 per ct. an’l prem. 

13 Days . 

. . . 13 per cent annual prem. 

165 Days. 

65 per ct. an’l prem. 

14 Days . 

. . . 13 per cent annual prem. 

180 Days or 6 mo. 

70 per ct. an’l prem. 

15 Days . 

. . . 14 per cent annual prem. 

195 Days. 

73 per ct. an’l prem. 

16 Days . 

. . . 14 per cent annual prem. 

210 Days or 7 mo. 

75 per ct. an’l prem. 

17 Days . 

. . . 15 per cent annual prem. 

225 Days. 

78 per ct. an’l prem. 

18 Days . 

. . . 16 per cent annual prem. 

240 Days or 8 mo. 

80 per ct. an’l prem. 

19 Days . 

. . . 16 per cent annual prem. 

255 Days. 

83 per ct. an’1 prem. 

20 Days . 

. . . 17 per cent annual prem. 

270 Days or 9 mo. 

85 per ct. an’l prem. 

25 Days . 

. . . 19 per cent annual prem. 

285 Days. 

88 per ct. an’l prem. 

30 Days . 

. . . 20 per cent annual prem. 

300 Days or 10 mo. 

90 per ct. an’l prem. 

35 Days . 

. . . 23 per cent annual prem. 

315 Days. 

93 per ct. an’l prem. 

40 Days . 

. . . 26 percent annual prem. 

330 Days or 11 mo. 

95 per ct. an’l prem. 

45 Days . 

. . 27 per cent annual prem. 

360 Days or 12 mo. 

100 per ct. an’l prem. 

50 Days . 

. . . 28 per cent annual prem. 





















358 


INSURANCE 


765. To find the cost of insurance at short rates, or the amount 
of premium to be returned if policy is canceled. 

1. Find the cost of insuring a stock of merchandise for $15,000 
for 3 mo. if the annual rate is |%. 

|% of $15,000 = $131.25, annual premium. 

40% of $131.25 = $52.50, premium for 3 mo. 

By the short rate scale, page 357, the premium for 3 mo. is 40% of the annual 
premium. The annual premium is $131.25; hence, the premium for 3 mo. is 
40% of $131.25, or $52.50. 

2. Property valued at $24,000 is insured for f of its value for 1 
yr. at 60j£ per $100. How much of the premium should be re¬ 
turned if the policy is canceled at the expiration of 8 mo. (a) by 
the insured? (6) by the insurance company? 

(а) f of $24,000 = $18,000. 

180 X $.60 = $108, annual premium. 

20% of $108 = $21.60, amount to be returned if insured 
cancels the policy. 

(б) 4 mo.: 12mo.::$? : $108. 

4 X $108 

-= $36, amount to be returned if insurer cancels 

^ the policy. 

(а) By the short rate scale the rate for 8 mo. is 80% of the annual rate. 
That is, the company will return 20% of the premium paid. The annual pre¬ 
mium is $108; hence, the amount returned is 20% of $108, or $21.60. 

(б) When the insurer cancels the policy, the amount returned bears the same 
ratio to the entire premium paid as the unexpired time bears to the entire time; 
hence, the proportion 4 mo. : 12 mo.:: $?: $108, the solution of which gives 
$36, the amount to be returned. 

PROBLEMS 

766. 1. A policy of insurance for $10,000 at f% per annum was 
dated Apr. 7, 1924. 5 mo. later it was canceled by the insured. 
How much of the premium was returned? 

2. May 10,1924,1 took out a policy of insurance on my furni¬ 
ture for $2400 at 40^ per $100 per annum. Feb. 19, 1925, I can¬ 
celed the policy. How much of the premium should be returned 
to me? 



FIRE INSURANCE 


359 


3. A stock of merchandise was insured for 1 yr. for $25,000 at 
|%. At the end of 7 mo. the insurance company canceled the 
policy. How much of the premium was returned? (Solve by 
proportion.) 

4 . A merchant insured his store and contents for 1 yr. for 
$16,500 at f%. At the end of 5 mo. and 15 da. the insurer canceled 
the policy. What was the return premium? (Solve by proportion). 

5. A policy of $32,000 at f% for 1 yr. was canceled by the in¬ 
sured at the expiration of 8 mo. Find the amount of the return 
premium. If the company had canceled the policy, what amount 
of premium would have been returned? 

6. The Aetna Insurance Company insured an office building for 
$45,000 at \% for 1 yr. Find the return premium if the policy 
was canceled at the end of 6 mo. (a) by the company; (6) by the 
insured. 

7 . Find the cost of insuring a stock of goods for $12,000 for 6 
mo. at 75^ per $100 per annum. 

REVIEW PROBLEMS 

767. 1. Find the cost of insuring a house for $8000 for 3 yr. 
under a 3-year policy if the annual premium is \%. 

2. At 60^ per $100 per annum, what is the cost of insuring a 
shipment of merchandise for $6500 for 30 da.? 

3. Property worth $75,000 is insured as follows: In the Hart¬ 
ford Insurance Company for $20,000 at 1|%, and in the Albany 
Insurance Company for $30,000 at 1J%. There is a $30,000 loss 
by fire during the first term of insurance. Find each company’s 
net loss under (a) an ordinary policy; (6) an average clause policy. 

Find the amount to be paid in each of the following cases, sup¬ 
posing each policy contains the Standard 80% Average Clause: 



Value of 
Property 

Face of 
Policy 

Damage 
by Fire 

Paid by 
Insurance Co. 

4 . 

$20,000 

$12,000 

$ 8,000 

? 

5 . 

28,000 

15,000 

15,000 

? 

6 . 

12,000 

5,000 

12,000 

? 

7 . 

16,500 

10,000 

3,000 

? 

8 . 

45,000 

20,000 

25,000 

? 












360 


INSURANCE 


9. An open policy of insurance was issued on merchandise stored, 
or to be stored, in a warehouse, the premium to be 80 £ per $100 per 
annum. Goods withdrawn inside of 1 yr. were to be charged the 
short rate. Find the total premium paid, and the total return 
premium on the following receipts and withdrawals: 


Receipts 


Withdrawals 


Date 

Amount 

Date 

Amount 

Apr. 

1,1924 

Silk, $ 8,400 

Oct. 1, 1924 

Woolens, S5000 

Apr. 

15,1924 

Furs, 12,800 

Oct. 10,1924 

Furs, 

9000 

May 

10,1924 

Woolens, 11,500 

Oct. 20,1924 

Silk, 

8400 

June 

1,1924 

Gloves, 6,500 

Nov. 10,1924 

Gloves, 

4500 

June 

20,1924 

Hosiery, 5,600 

Dec. 1,1924 

Woolens, 

6500 


LIFE INSURANCE 

768. Life insurance is a contract in which the insurer, for a con¬ 
sideration, agrees to pay a specified sum of money at the death of 
the insured, or at some fixed time. 

769. The consideration is called the premium and may be paid 
in one sum at the beginning of the contract, or in annual payments 
in advance for a series of years. 

770. Life insurance policies may generally be classified as fol¬ 
lows: (1) whole life; (2) term; (3) endowment. 

771. Whole life policies are those in which the sum insured is 
payable at death only. The premiums may be paid annually 
during the life of the insured, or may be limited to 10, 15, 20, or 25 
annual payments. In the latter case the policy becomes paid up 
for life, after the specified number of payments have been made. 

772. In term policies, the sum insured is payable only in the 
event of death during a fixed term. At the expiration of the term 
the insurance ceases. 

773. Endowment policies provide for the payment of the sum in¬ 
sured at a fixed date if the insured is then living; or, in case of death 
before that date, the amount of the policy is paid to the beneficiary. 

774. The beneficiary is the person named in the policy to receive 
payment upon the death of the insured. 











LIFE INSURANCE 


361 


775. A portion of every premium is required by law to be set 
aside and properly invested for the purpose of providing a reserve 
fund from which death losses and maturing policies are to be paid. 

776. If the insured discontinued his policy he is entitled to the 
reserve set aside to meet his policy when it matures. He may re¬ 
ceive the cash value of the reserve, or he may take a paid-up policy 
for the sum which the reserve will purchase or he may use the 
reserve to extend the time of the policy for such a time as the re¬ 
serve will purchase. 

777. The following table of values shows the options and their 
value, in one insurance company to which the insured is entitled 
in case of discontinuance of the payment of premiums. 

Table of Values* 


At End 

op Year 

Loan or Cash 

Surrender Value 

In Case of Lapse of Policy 

Paid-up Insurance 
on Surrender 

Or Extended Insurance without 
any Notice from the Insured 

3d 

$ 25.96 

$ 150 

5 years 

40 days 

4th 

39.05 

200 

7 

14 

5th 

52.71 

250 

9 

30 

6th 

67.00 

300 

11 

112 

7th 

81.92 

350 

13 

250 

8th 

97.52 

400 

16 

51 

9th 

113.83 

450 

18 

184 

10th 

130.88 

500 

20 

261 

11th 

148.71 

550 

22 

279 

12th 

167.37 

600 

24 

199 

13th 

186.89 

650 

26 

66 

14th 

207.33 

700 

27 

248 

15th 

228.75 

750 

29 

27 

16th 

251.18 

800 

30 

144 

17th 

274.70 

850 

31 

308 

18th 

299.37 

900 

33 

135 

19th 

325.25 

950 

35 

31 

20th 

352.42 

1000 Policy 

Fully Paid 


25th 

402.60 




30th 

458.53 





*Age of insured 22 yr. Annual premium, $25.68 per $1000, payable for 20 
yr. Amount of the policy payable at the death of the insured. 















\ 


362 INSURANCE 

778. The following tables show the annual premium on $1000 
of insurance on different kinds of policies and for different ages of 
the insured in a leading insurance company. 


RATES FOR $1000 INSURANCE 

Whole Life Policies, with Premiums payable 
annually during Life, or in 10 or 20 Years, or 
by a Single Payment. 

RATES FOR $1000 INSURANCE 

Endowment Insurance Policies, pay¬ 
able at the Ertd of the Terms Stated, 
or on Prior Death; Annual Premiums. 

Age 

Annual 

Pre¬ 

miums 

Ten 

Annual 

Pre¬ 

miums 

Twenty 

Annual 

Premiums 

Single 

Payment 

Age 

10 

Years 

15 

Years 

20 

Years 

25 

Years 

20 

$18.00 

$46.75 

$27.76 

$372.54 

20 

$106.30 

$67.79 

$48.92 

$37.92 

21 

18.40 

47.43 

28.17 

377.36 

21 

106.34 

67.83 

48.07 

37.98 

22 

18.80 

48.13 

28.60 

382.33 

22 

106.37 

67.88 

49.03 

38.05 

23 

19.23 

48.86 

29.04 

387.46 

23 

106.41 

67.92 

49.08 

38.12 

24 

19.67 

49.60 

29.50 

392.74 

24 

106.45 

67.97 

49.14 

38.20 

25 

20.14 

50.38 

29.98 

398.20 

25 

106.49 

68.02 

49.21 

38.28 

26 

20.63 

51.18 

30.47 

403.83 

26 

106.53 

68.08 

49.28 

38.38 

27 

21.15 

52.00 

30.98 

409.63 

27 

106.58 

68.14 

49.36 

38.48 

28 

21.69 

52.86 

31.51 

415.61 

28 

106.63 

68.21 

49.45 

38.59 

29 

22.26 

53.74 

32.06 

421.78 

29 

106.69 

68.28 

49.54 

38.71 

30 

22.85 

54.65 

32.62 

428.14 

30 

106.75 

68.36 

49.64 

38.85 

31 

23.48 

55.59 

33.21 

434.70 

31 

106.82 

68.45 

.49.76 

39.00 

32 

24.14 

56.56 

33.83 

441.45 

32 

106.90 

68.55 

49.89 

39.18 

33 

24.84 

57.56 

34.47 

448.41 

33 

106.98 

68.65 

50.03 

39.37 

34 

25.58 

58.60 

35.13 

455.57 

34 

107.06 

68.77 

50.18 

39.58 

35 

26.35 

59.67 

35.82 

462.95 

35 

107.16 

68.90 

50.36 

39.82 

36 

27.17 

60.78 

36.54 

470.54 

36 

107.27 

69.04 

50.56 

40.09 

37 

28.04 

61.92 

37.30 

478.36 

37 

107.39 

69.20 

50.78 

40.39 

38 

28.95 

63.11 

38.08 

486.39 

38 

107.52 

69.39 

51.03 

40.72 

39 

29.92 

64.33 

38.91 

494.65 

39 

107.67 

69.59 

51.30 

41.10 


EXAMPLES 

779. 1. What is the annual cost of a whole life policy of $5000, 
age 30 yr. if payments are to be made during the life of the in¬ 
sured? 

In the table showing rates on whole life policies, in the first money column 
opposite age 30 is the amount, $22.85, the rate on $1000. On $5000, the cost 
is 5 X $22.85, or $114.25. 

2. Find the annual cost of a 20-yr. endowment policy of $3000, 
age 25. 

In the table showing rates on endowment policies, in the column headed 
“20 years” opposite age 25, is found $49.21, the annual cost of $1000. $3000 

costs 3 X $49.21, or $147.63. 



















LIFE INSURANCE 


363 


3 . A 15-yr. endowment policy of $15,000 was taken by a man 35 
yr. of age. If he lives till the policy matures, how much less will 
he receive from the insurance company than he would have had if 
had placed his annual premiums in a savings bank and received 
4%, interest compounded annually? 

The annual premium is 15 X $68.90, or $1033.50. 

A payment of $1 each year, for 15 yr., will amount to $20.824531 (see page 
333), if compounded annually at 4%. $1033.50 will amount to 1033.50 X 
$20.824531, or $21,522.15, if deposited in savings bank. $21,522.15 — $15,000 
= $6522.15 less from the insurance company. 

4 . If the insured in example 3 had died at the age of 40 yr., how 
much more would the beneficiary have received from the company 
than the amount of the annual premiums at 4% compound in¬ 
terest? 

Each annual premium is $1033.50, and has been paid for 5 yr. At 4% com¬ 
pound interest the amount would be 1033.50 X $5.632975 (see page 333), or 
$5821.68. 

$15,000 — $5821.68 = $9178.32, excess received from the company. 

PROBLEMS 

780. 1 . Find the annual cost of a $5000 policy on the life .of a 
person 39 yr. of age if premiums are to be paid during life. 

2. At age 39, a person’s expectation of life is 25 yr. If the person 
insured in problem 1, lives and pays the premium during his ex¬ 
pectation of life, what sum will he actually have paid to the in¬ 
surance company? At 4% compound interest the amount of his 
payments exceed by how much the face value of the policy? 

3. A man 22 yr. of age takes out a whole life policy of $5000 
with premiums payable annually for 20 yr. At the age of 35 he 
ceases making payments. What is the surrender value of his 
policy? What amount of paid-up insurance has he to his credit? 
For what length of time will insurance for the face value of the 
policy be extended if he so chooses? (See page 361.) 

4. A man takes a 15-yr. endowment policy of $20,000 on his 
life at the age of 30 yr. If he survives the 15-yr. period, how 
much more will he have paid the company than the face of his 
policy? How much would his payments have amounted to at 3% 
compound interest? 


SAVINGS BANKS 


781. A savings bank is a bank under the control of state laws, for 
the purpose of receiving small deposits of money and paying interest 
thereon. 

782. Interest is usually paid on any number of dollars from $1 to 
$3000, on condition that such sum has been on deposit for an entire 
interest term. 

783. The interest term is the period of time between the dates on 
which interest payments are due. 

784. If interest payments are due Jan. 1 and July 1, the inter¬ 
est term is 6 mo. If due Jan. 1, Apr. 1, July 1, and Oct. 1, 
the interest term is 3 mo., etc. In some states interest begins to 
accrue from the first of each calendar month on all sums on deposit 
at that time. A withdrawal of funds cancels the interest on the 
sum withdrawn for the entire interest term. 

785. When interest becomes due it may be drawn out of the 
bank or it will be placed to the credit of the depositor, in which 
case it draws interest the same as the other deposits. Savings 
banks, therefore, pay compound interest. 

786. Each depositor receives a small pass book in which are 
entered all deposits and withdrawal Checks cannot be drawn 
against a savings bank account. 

787. To find the balance due on a savings bank account. 

Find the balance due July 1, 1924, on the following: The 
account was opened Sept. 30, 1922, by depositing $50. Dec. 31, 
1922, deposited $20; Apr. 8, 1923, withdrew $25; June 20, 1923, 
deposited $75; July 8, 1923, withdrew $30; Dec. 20, 1923, depos¬ 
ited $100; June 5, 1924, withdrew $40. Interest at 4% per 
annum is allowed from the first of each quarter, and is credited Jan. 
1 and July 1. 


364 


SAVINGS BANKS 


365 


Drafts 

Deposits 

Date 

Balance 


$ 50.00 

1922 

Sept. 30 

$ 50.00 


20.00 

Dec. 31 

70.00 


Int. .50 

1923 

Jan. 1 

70.50 

$25.00 


Apr. 8 

45.50 


75.00 

June 20 

120.50 


Int. .90 

July 1 

121.40 

30.00 


July 8 

91.40 


100.00 

Dec. 20 

191.40 


Int. 1.82 

1924 

Jan. 1 

193.22 

40.00 


June 5 

153.22 


Int. 3.06 

July 1 

156.28 


The simplest method of solution is to use a form like the above, and to enter 
the deposits, withdrawals, and interest credits in their proper chronological 
order. 

Since interest is allowed from the first of each quarter, the original deposit of 
$50 draws interest from Oct. 1, 1922 to Jan. 1, 1923,—3 mo. Interest is not 
allowed on the deposit of $20 made Dec. 31, 1922, because it has not been on 
deposit during the quarter. The interest on $50 for 3 mo. is 50 i which 
makes a total amount on deposit of $70.50. On Apr. 8, 1923, a withdrawal of 
$25 reduces the blance to $45.50. The deposit made June 20 increases the 
balance to $120.56, but draws no interest until after July 1,1924. The amount 
on which to reckon interest for the interest term from Jan. 1, 1923, to July 1, 
1923, is $45 (interest is not paid on parts of a dollar), the smallest balance dur¬ 
ing the term. The interest is 90 ^ which makes the balance $121.40. The 
draft of $30, July 8, 1923, leaves a balance of $91.40. The deposit Dec. 20, 
1923, of $100 increases the balance to $191.40. The interest on $91, the 
smallest balance of the term, is $1.82, making the balance Jan. 1,1924, $193.22. 
The withdrawal of $40 June 5, 1924, leaves $153.22 on deposit. The interest 
on $153 for 6 mo. is $3.06, making a balance due of $156.28. 

PROBLEMS 

788 o Assuming that interest at 4% per annum is allowed from 
the beginning of each quarter, and is credited Jan. 1 and July 1, 
prepare a statement similar to the illustration on this page for each 
of the following savings bank accounts: 












366 


SAVINGS BANK 


1. Deposited Apr. 1, 1922, $125; May 15, 1922, $75; withdrew 
July 3, 1922, $25; deposited, Oct. 1, 1922, $50; deposited Dec. 28, 
1922, $100; withdrew June 1, 1923, $60; deposited Aug. 30, 1923, 
$125; deposited Mar. 30, 1924, $200. How much is due July 1, 
1924? 

2. Balance Jan. 1, 1922, $387.50; deposited Oct. 1, 1922, $225; 
withdrew May 1, 1923, $100; deposited Apr. 1, 1924, $300; with¬ 
drew Oct. 20, 1924, $150. How much was due Jan. 1, 4925? 

3. Balance Oct. 1, 1922, $575; withdrew Jan. 4, 1923, $150; de¬ 
posited July 1, 1923, $400; deposited Sept* 15, 1923, $200; with¬ 
drew Nov. 10, 1923, $50; deposited Feb. 18, 1924, $500. Find 
balance due July 1, 1924. 

4-6. Find the balance due on problems 1-3 if interest is allowed 
from the first of each month, credit being given Jan. 1 4nd July 1. 

POSTAL SAVINGS BANKS 

789. The postal savings system was established in the United 
States January 3, 1911, at 48 second-class post offices, one in each 
state and territory. 

790. Deposits are evidenced by postal-savings certificates issued 
in fixed denominations of $1, $2, $5, $10, $20, $50, $100, $200, and 
$500, each bearing the name of the depositor, the number of his ac¬ 
count, the date of issue, the name of the depository office, and the 
date on which interest begins. 

791 . Interest is allowed on all deposits at the rate of 2 per cent 
per annum, computed on each savings certificate separately, and 
payable annually. No interest is paid on money which remains 
on deposit for a fraction of a year only. 

792. Deposits bear interest from the first day of the month next 
following that in which deposited. 

793. Compound interest is not allowed on an outstanding certi¬ 
ficate, but a depositor may withdraw interest payable and include 
it in a new deposit, which bears interest at the regular rate. 

794. A depositor is permitted to exchange his deposits for reg¬ 
istered or coupon United States postal savings bonds, issued in de¬ 
nominations of $20, $100, and $500, bearing interest at the rate 


SPEED TEST 


367 


of 2J per cent per annum, payable semiannually. The exchange 
may be made under date of Jan. 1 and July 1 of each year. 

PROBLEMS 

795. l. On Feb. 28, 1911 (after 2 months’ operation), the num¬ 
ber of banks was 48, the total number of depositors was 3664, and 
the amount on deposit was $133,869. Dec. 30, 1911, there were 
5132 depositories, 162,697 depositors, and $10,614,676 on deposit. 
Show the per cent of increase in the number of depositories and of 
depositors, and in the amount of deposits as compared with Feb. 
28, 1911. 

2. A stenographer bought $15 worth of postal savings certifi¬ 
cates each month for three years, beginning Dec. 20, 1920. She 
drew her interest as it became due. On what date was she first 
able to draw interest? How much did she receive at that time? 
Prepare a table showing the amount of interest she could draw each 
month to and including January 1, 1924. 

3. Beginning May 20,1920, a man made a monthly deposit of $25 
in a postal savings bank. On Jan. 1, 1922, he exchanged his total 
deposits for a postal savings bond. Assuming that he made no 
deposits after Jam 1, 1922, and that he drew his interest as it be¬ 
came due, how much interest did he receive on his deposits and 
bond to and including July 1, 1924? 

4. On the last business day of each month, a man deposits $10 
in a postal savings bank, for five years. If no interest has been 
drawn, what amount will he then have to his credit? 

EXAMINATIONS 
SPEED TEST 

796. Minimum time thirty minutes; maximum, one hour. De¬ 
duct one credit for each minute beyond minimum time. 

1. Find the interest in the 2. Find the proceeds of the 


following cases: 


following discounted notes: 

$500 for 2 yr. 2 mo. 

6 da. at 6%. 

$350 

90 da. 

6%. 

$300 for 5 yr. 6 mo. 18 da. at 6%. 

$750 

75 da. 

6%. 

$930 for 6 yr. 8 mo. 

at 5%. 

$960 

45 da. 

4%. 

$750 for 4 yr. 6 mo. 

at 31%. 

$150 

72 da. 

5%. 


368 


SPEED TEST 


3. Find balance due July 1, 
1924, on the following (United 
States Rule): 

Face of note, $3000; date, July 1, 
1923; Interest, 6%. 

Payments, Nov. 1, 1923, $1000. 

Apr. 1, 1924, $1500. 

5. Find the present worth of 
each of the following notes: 

$515 due in 6 mo. at 6%. 
$2020 due in 72 da. at 5%. 

$ 858.50 due in 60 da. at 6%. 
$1540.25 due in 90 da. at 4%. 

7. Find the total tax on the 
following: 

Farm valued at $2400 at .0175. 

House and lot valued at $18,000 at 
01875 

City lot valued at $5500 at $1.60 per 
$ 100 . 

Store valued at $30,000 at $1.7246 per 
$ 100 . 


4. Find the accurate interest 
in the following cases: 

$3650 for 125 da. at 5%. 

$7300 for 116 da. at 6%. 

$1095 for 17 da. at 4%. 

$ 550 for 146 da. at 3£%. 

6. Find the total insurance 
on the following properties: 
Store worth $25,000 at #% premium. 
Mdse, worth $40,000 at 11 % premium. 
House worth $15,000 at 60 j: per $100 
premium. 

Furniture worth $5000 at 80^ per 
$100 premium. 

8. At what price each shall 
these articles be sold to gain the 
rates given? 


Cost per Dozen 

Rate of Gain 

$18 

20% 

$24 

40% 

$15 

33i% 

$14.40 

25% 


9-10. Add vertically and horizontally and find grand total: 


16832 

21265 

15293 

17294 

13846 

17286 

16450 

18642 

8287 

18643 


17234 

18875 

16872 

5570 

9237 

18765 

27436 

18778 

21299 

17694 


WRITTEN TEST 


797. l. In a certain county containing 25,482 taxable inhabit¬ 
ants, a tax of $103,294.60 is assessed for town, county, and state 
purposes: a part of this sum is raised by a tax of 30^ on each 
poll; the entire valuation of the property on the assessment roll is 
$38,260,000. What is the per cent tax, and what is a person’s 
tax who pays for 1 poll and whose property is valued at $9470? 


WRITTEN TEST 


369 


2. Find the interest on $738.72 for 1 yr. 3 mo. 22 da., at 5%. 

3 . I bought a horse for $360 and sold it for a note at 60 da. 
for $418. I discounted the note at 6% the day it was made. 
What was my gain? 

4 . A store was insured for $12,000 at f% and the goods for 
$15,000 at 1J%. What was the entire premium? 

5. A wholesale grocer sold 200 bbl. of molasses, each containing 
31 gal. 2 qt. at $.60 a gallon, receiving in payment a note at 90 da., 
with interest at 6%. What would be the proceeds of this note if 
discounted the day it was written at a bank at 7\% per annum? 

6. For the installation of a system of city waterworks a city 
issues bonds to the amount of $1,500,000 payable in 20 yr. What 
amount must he set aside each year as a sinking fund at 4% 
compound interest to pay the bonds when they are due? 

7. A man deposited $15,000 in a bank June 1, 1924, .receiving 
therefore a certificate of deposit which stated that 3% interest 
would be paid if the money was left on deposit 6 mo. or more. The 
bank loaned the money at 5%. Dec. 20, 1924, the man asked for 
his money and interest. If the bank pays exact interest and 
charges ordinary interest, find the gain to the bank, the money 
having been loaned all the time. 

8. A merchant bought a bill of goods invoiced at $5475.76, less 
10%"and 5%, terms net 60 da. or 5% 10 da. Not having the 
ready money, he borrowed it at 6% interest and accepted the 10-da. 
terms. Find his gain by so doing. 

9. On a note of $4000 dated June 1, 1923, and drawing interest. 
at 6%, the following payments were made: Sept. 16, 1923, $1000; 
Jan. 19, 1924, $1500; and Apr. 1, 1924, $1200. How much was 
required to settle the balance Aug. 1, 1924? 

10. A factory was insured under a policy containing the standard 
eighty-per-cent-average clause, as follows: Building for $22,000 at 
1}%; machinery for $25,000 at lf%; and the stock at $15,000 at 
1 f%. The building is worth $30,000, the machinery $35,000, and 
the stock $20,000. Fire broke out causing a damage as follows: 
to the building, $8000; machinery, $6000; and to the stock, 
$12,000. Find the amounts paid by the company. 

VAN TUYL’S NEW COMP. AR.—24 


STOCKS 


798. Stock, or capital stock, is the name given to the capital of a 
corporation. 

799. A corporation is an imaginary person composed of several 
real persons. It is formed only by the consent of the state, and 
hence, has only those rights and powers that are granted to it by 
the state as set forth in its charter. The list of powers, rights, and 
duties are stated in writing and issued to the corporation. This 
written instrument is called a charter. 

800. The capital stock of a corporation is divided into shares, 
usually $100 each, though shares may be of various values. 
Reading Railroad shares are $50 par value. Mining stocks are at 
all prices from 10 £ to $100. 

801 . A stockholder is any person who owns one or more shares of 
stock. As evidence of ownership of stock, each stockholder re¬ 
ceives a written statement called a stock certificate, showing the 
number of shares he owns and their par value. 

802. Stock is either common or preferred. Common stock is 
the kind most usually issued by a corporation, and carries no 
guaranty of a specified dividend. Preferred stock is stock on 
which dividends are due and payable before dividends may be 
paid on common stock. Preferred stock is often issued in times 
of emergency to provide working capital or to pay debts. 

803. The par value of stock is the value stated on the stock cer¬ 
tificate. The market value is the price for which the stock can be 
sold. 

804. The market value of stock depends chiefly upon the rate of 
dividends paid or earned by the company. If the dividends are 
small, the market value is below par (less than face value, or at a 
discount), and if the dividends are large, the market value is above 
par (or at a premium). 


370 


STOCKS 


371 


805. Illustration of a Stock Certificate 



806. Shares of stock are transferred by assignment. The 
assignee, then, has all the rights and privileges of the original 
stockholder. 


807. Form of Assignment 























372 


STOCKS 


808. The following prices of certain stocks were printed in the 
New York Times , July, 1923. 

New York Stock Exchange 


Stock Exchange Sales and Quotations July 17, 1923 


Sales 

First 

High 

Low 

Last 

Net 

Change 

800 Am. Car & Foundry . . 

151* 

153 

151* 

153 

+ 1 

200 Am. Cotton Oil. 

4* 

4* 

4* 

4* 

- - - - 

1,900 Am. Smelting & Ref. . . 

57* 

57* 

56* 

56* 

-1* 

400 Am. Sugar Refining . . . 

62f 

62f 

62* 

62* 

" * 

900 Am. Tel. & Tel. 

122| 

122| 

122| 

122| 

_ 

1,900 Atch. Top. & S. F. . . . 

98* 

99* 

98* 

99* 

_ 

300 Ches. & Ohio. 

59 

59 

59 

59 

-1 

400 Chi. Mil. & St. Paul . . 

18* 

19* 

18f 

181 

- - - - 

300 Chi. & N. W. 

70f 

70* 

70 

70* 

+ * 

100 Colorado Southern . . . 

28 

28 

28 

28 


1,600 Consolidated Gas . . . 

59f 

60 

59f 

60 

+ i 

700 Delaware & Hudson . . 

105 

105 

104* 

1041 

_ i 

4 

500 Eastman Kodak, pfd . . 

108* 

108* 

1071 

108 

“ t 

400 Erie. 

m 

Hi 

11* 

11* 

+ 1 

1,100 Erie, 1st pfd. 

18* 

18f 

18* 

181 

+ 8 

100 Illinois Central. 

109 

109 

109 

109 

+ 1 

500 Lehigh Valley. 

59* 

59* 

59 

59* 

+ * 

300 Maxwell Motors Cl. A . 

39 

39 

38* 

38* 

-1 

1,200 Missouri Pacific, pfd. . • 

30| 

31* 

30| 

31* 

+ t 

400 National Lead. 

118 

118 

117 

117 

"If 

4,800 New York Central . . . 

971 

98* 

97 

971 


2,100 Northern Pacific .... 

651 

66* 

64| 

66* 

+ t 

100 Otis Elevator. 

116 

116 

116 

116 

+ * 

1,200 Pennsylvania R. R. . . . 

43* 

43* 

43 f 

431 

_ 1 

4 

1,100 Reading. 

71 

71f 

70f 

71* 

_ 

1,100 Southern Pacific .... 

861 

86 f 

86* 

86* 

- * 

76,100 Studebaker Co. 

102* 

103* 

101* 

103* 

+ I 

100 United Fruit. 

166 

166 

166 

166 

+ 1 

17,100 U. S. Steel. 

90| 

90* 

89f 

90* 


500 U. S. Steel, pfd. 

117* 

118 

117* 

118 


1,000 Western Union Tel. . . . 

104 

106* 

104 

106* 

+2* 

1,000 Woolworth (F. W.) Co. . 

234 

236f 

234 

2361 

+6* 


809. The profits of a corporation, after deductions are made for 
sinking fund, reserve for bad debts, depreciation, etc., may or 
may not be divided among the stockholders according to the 





















STOCKS 


373 


decision of the board of directors. Such profits, when voted by 
the directors, are called dividends. Dividends are reckoned as 
a per cent of the par value of the issued stock of the company. 
Dividends on preferred stock (if any) are voted first. The balance 
of the profits may be voted as dividends on the common stock. 

Note. In some cases, however, the preferred stockholders share equally 
with the common stockholders if the profits are sufficient to pay on all stock 
more than the rate on the preferred stock. 

810 . An assessment is a certain per cent of the par value of 
the stock levied against the stockholders to pay losses, etc. 

811 . Stocks are bought and sold by stockbrokers who make a 
business of buying and selling stocks for others. For their services 
they charge a commission, or brokerage. The rate charged differs 
on the different exchanges. On some exchanges it is \% of the 
par value of the stock bought or sold. On the New York Stock 
Exchange the rate of commission for buying and selling stocks is as 
follows: 

On stock selling below $10 a share the rate is $.07| a share. 

On stock selling at $10 a share and above, but under $125 a share the 
rate is $.15 a share. 

On stock selling at $125 a share and above, the rate is $.20 a share. 

The minimum commission charge on a transaction is $1.00. 

812 . Stock quotations on all exchanges, except Montreal and 
Toronto, are in dollars (or cents) per share. Thus Illinois Central 
(see page 372) is quoted at 109, which means $109 per share. 

813 . There is a Federal tax of $2 per 100 shares of stock of $100 
par value on all sales of stock, the tax in all cases to be paid by the 
seller. In New York State there is a similar tax of $2 per 100 
shares on all sales of stock. This tax is likewise paid by the seller. 
If the par value of the stock is other than $100 the tax bears the 
same ratio to $2 per 100 shares that the par value of the stock bears 
to $100. Thus if a sale of 100 shares of stock having a $50 par 
value is made, the tax on such sale is $1 per 100 shares, because 
the par value of the stock is only one half of $100. 

EXPLANATION OF STOCK EXCHANGE TERMS 

814. 1. Bear. A broker who operates for declining prices. He has sold 
“short” (see 5) and wishes to cover his short sale at a lower price. 


374 


STOCKS 


2. Bull. A broker who operates for rising prices. He has bought stocks 
(is “long”—see 4) and wishes to sell them at higher prices. 

3. Collaterals. Stocks or bonds deposited as security for a loan. 

4. Long. A trader who buys stocks for a rise in price is long. 

5. Short. When a trader has sold more stock than he owns, he is said to 
be short. 


815. To find the cost of a purchase, or the proceeds of a sale, 
of stock. 

Unless otherwise indicated, brokerage at the rates charged on 
the New York Stock Exchange, and the Federal transfer tax of $2 
per hundred shares will be reckoned on problems in stocks in this 
text. 

Examples. 

1. Find the cost of 1250 shares of American Cotton Oil at the 
lowest price quoted on page 372. 

$4.25 + $.07J = $4.32|, gross cost of 1 share. 

1250 X $4.32J = $5406.25, cost. 

2. Find the proceeds of a sale of 400 shares U. S. Steel, pfd. 
at the highest price quoted on page 372. 

400 X $118 = $47,200, gross proceeds of sale. 

400 X $.15 = $60, commission. 

4.00 = $2 = $8, transfer tax. 

$47,200 — ($60 + $8) = $47,132, proceeds of sale. 


816. Using the quotations on page 372, making allowance for 
brokerage, and for Federal transfer tax on all sales, find the 
results as indicated in the following: 


cost of: 

6. 325 Lehigh Valley 

7. 560 Nat. Lead 

8. 750 New York Central 

9. 600 Northern Pacific 
io. 800 U. S. Steel 


Find, at the lowest price, the 
l. 350 Am. Smelting & Ref. 

2- 125 Am. Tel. & Tel. 

3. 375 Chi. & N. W. 

4. 175 Consol. Gas. 

5. 225 Del & Hudson 

Find, at the highest price, the 

11. 200 Am. Car & Foundry 

12. 275 Studebaker Co. 

13. 175 Missouri Pacific, pfd. 


proceeds of: 

14. 450 Reading (Par $50) 

15. 375 United Fruit 

16. 650 Otis Elevator 


STOCKS 


375 


17. 500 F. W. Woolworth & Co. 19. 1250 Western Union Tel. 

18. 475 Northern Pacific 20. 1375 Erie 


Find the gain or the loss on the following stocks if bought at the 
lowest and sold at the highest price quoted on page 372, allowing 
for brokerage each way: 


21. 150 Am. Car and Foundry 

22. 200 Atch., Top. & S. F. 

23. 300 Chi., Mil. & St. P. 

24. 450 Lehigh Valley (Par $50) 

25. 750 Erie, 1st pfd. 


26. 575 Eastman Kodak, pfd. 

27. 875 F. W. Woolworth & Co. 

28. 500 Maxwell Motors 

29. 315 Illinois Central 

30. 625 Penn. R.R. (Par $50) 
817. To find the number of shares. 

How many shares of Maxwell Motors, at the last price quoted 
on page 372, including brokerage, can be bought for $9276? 
$38.50 + $.15 = $38.65, gross cost of 1 share. 

$9276 -7- $38.65 = 240, no. shares. 

The gross cost of 1 share is the sum of the price and the brokerage. Divid¬ 
ing the total investment by the cost of 1 share gives the number of shares. 

For prices of stocks and brokerage charges, see pages 372 and 
373. 

PROBLEMS 


Use the “first” price in these problems. 

818. l. How many shares of Am. Sugar Refining stock can be 
bought for $11,048.40? 

2. An investment of $7296 will buy how many shares of Erie? 

3. A man sold 220 shares of Eastman Kodak, pfd., and invested 
the proceeds in Delaware & Hudson. How many shares did he 
buy, and how much cash had he left? (Reckon in whole shares.) 

4. To provide himself with $27,500 a man sold 175 National 
Lead, and a sufficient quantity of Otis Elevator for the balance of 
the funds needed. How many shares of Otis Elevator did he sell, 
and how much additional money over and above the $27,500 
did he receive? 

5. A broker bought on his own account 1000 shares of F. W. 
Woolworth & Company at the first price, and when it advanced 
to the highest price (page 372) he sold enough of the stock to make 
a profit of $1638. How many shares did he sell? 


376 


STOCKS 


819. To find the dividend on stocks. 

1. A man invested $42,476 in Am. Car and Foundry at the first 
price. If the stock paid 12% dividends, how much was his annual 
income? 

$151.50 + $.20 = $151.70, gross cost of 1 share. 

$42,476 -5- $151.70 = 280, no. shares. 

Value of 280 shares = $28,000, par value of stock. 

12% of $28,000 = $3360, annual income. 

Since the dividends are always based on the par value of the stock, it is 
necessary, first, to find the par value of the stock. The number of shares is 
found as in Art. 817. At $100 a share, the par value of the stock is 280 times, 
$100, or $28,000. 12% of the par value gives the annual income, $3360. 

2. The capital stock of a corporation is $5,000,000. The gross 
earnings are $1,250,000; the operating expenses are $900,000. 
$50,000 is carried to the surplus fund. What per cent dividend 
can be declared? If A owns 250 shares, how much will he receive? 

$900,000 + $50,000 = $950,000, to deduct from gross earnings. 

$1,250,000 — $950,000 = $300,000, total dividend. 

$300,000 -r- $5,000,000 = .06 = 6%, rate of dividend. 

Value of 250 shares = $25,000, par value of A’s stock. 

6% of $25,000 = $1500, A’s dividend. 

From the gross earnings there are deducted the operating expenses, and the 
amount set aside for the surplus fund, which leaves $300,000 as the total 
dividend. Dividing by the capital stock, $5,000,000, gives 6%, the rate of 
dividend. The par value of A’s stock is $25,000. 6% of $25,000 gives $1500, 
A’s dividend. 

PROBLEMS 

820. 1. An investment of $63,234 was made through a broker 
in F. W. Woolworth & Company at the low price. The stock 
paid 8% dividends. Find the income from the investment. 

2. A man drew $16,682 from a bank in which he was receiving 
3|% interest annually, and invested it in Pennsylvania Railroad 
stock at the low price. If the stock paid 3% dividends, was his 
income increased or diminished, and how much per annum? 

3 Which is better, and how much per annum, to invest $33,240 
in United Fruit at the low price, paying 8% dividends, or to de¬ 
posit it in savings banks paying 4% compounded semiannually? 


STOCKS 


377 


4. A railroad company has a capital stock of $42,400,000. The 
gross income for 1 yr. was $44,829,555.20; the total expenses were 
$38,363,381.79. Find the largest whole per cent dividend that 
could be declared, and the balance left over for the reserve fund. 
What amount of dividend would be received by a man who owned 
1500 shares of the stock? 

821. To find the investment required to produce a given income. 

How much must be invested in United States Steel, pfd. at the 
last price (page 372) to yield an annual income of $1750 if the stock 
pays 7%? 

7% of $100 = $7, income from 1 share. 

$1750 4- $7 = 250, no. of shares. 

250 X $118.15 = $29,537.50, investment. 

7% stock pays 7% of its par value. 7% of SI00 is $7, income from 1 share. 
To produce S1750 requires 250 shares. 1 share costs $118.15 (including 
brokerage). 250 shares cost $29,537.50. 

822. 1 . What investment in New York Central stock paying 
7% dividends, at the low price, will yield an income of $2450? 

2. When the Northern Pacific pays 5% dividends, how much 
must be invested in its stock at the first price to yield $2250 
annually? 

3. An investment in Illinois Central stock paying 7% at the last 
price pays $2625 annually. Find the amount of the investment. 

4. You desire an income of $3500. You have an option on New 
York Central 7% stock, and on Northern Pacific 5% stock, at 
the low prices quoted on page 372. Which stock requires the 
smaller investment? 

823. To find what per cent an investment pays. 

When Western Union Telegraph stock at the low price pays 7% 
dividends, what rate per cent does the stock pay on the invest¬ 
ment? 

7% of $100 = $7, income from 1 share. 

$104 + $.15 = $104.15, cost of 1 share. 

$7 4- $104.15 = .0672; that is 6.72% on the investment. 

Since the income from 1 share is $7, and the cost of 1 share is $104.15, the 
rate of income is found by dividing the income by the cost, which gives 6.72%. 


378 


STOCKS 


824. l. If 6% stock is bought at 120, no brokerage, what per 
cent on the investment does the stock pay? 

2. When a stock costs 210, and pays quarterly dividends of 3%, 
what rate does it pay on the investment (no brokerage)? 

3. Which is better; and how much per cent, to invest, at the 
last price, in Pennsylvania Railroad stock paying 3% dividends, 
or in Consolidated Gas stock, paying 5% dividends? 

4. You are offered American Telephone and Telegraph, stock, 
paying 9% dividends, or New York Central stock, paying 7% 
dividends. Reckoning at the low price, which is the better invest¬ 
ment, and how much per cent better? 

5. A broker owns 600 shares of U. S. Steel pfd., which pays 7% 
dividends. He sells it at 120 and buys with the proceeds Atchison, 
Topeka and Santa Fe at 96. If the Atchison, Topeka and Santa 
Fe stock pays 6% dividends, is his income increased or diminished, 
and what per cent? (Make no allowance for transfer tax.) 

825. To find the investment to yield a given rate per cent. 

At what price will a 4% stock pay 5% on the investment? 

4% of $100 = $4, income from 1 share. 

$4 -f- .05 = $80, cost of 1 share. 

Since this is to be a 5% investment, the dividend, $4, must be equal to 5% 
of the cost of the share. Hence, $4 -r- .05 gives $80, the cost of 1 share. 

826. 1. A man desires a 5% investment. What price can he 
afford to pay for 6% stock? 

2. A 7% stock pays only 4% on the investment. What is the 
market value, brokerage J%? 

3. What is the market value of stock paying dividends of 4J%, 
if an investment in them, including brokerage, pays 5%? 

4. Find the cost, including brokerage, of 350 shares of stock pay¬ 
ing 6% dividends, if the rate on the investment is 4|%. 

5. A broker has an option on 500 shares of 5% stock at a price 
to yield 8% on the investment, or a sufficient quantity of stock pay¬ 
ing 2% semiannually at 50, to require the same investment. He 
accepts the latter stock. Is his income more or less, and how 
much, than it would have been on the other stock, if money is 
worth 6% per annum? 


STOCKS 


379 


PROBLEMS 

1. A man purchased 600 shares of stock paying annual dividends 
of 8% at $227.50 a share, brokerage \%. What is the rate o 
interest on the investment? 

2. A corporation has a capital stock of $250,000 divided into 
1500 shares of common stock and 1000 shares of preferred stock, 
the latter paying 7% dividends. In 1923, the company earned a 
net profit of $29,250. What is the largest whole per cent dividend 
that can be declared on the common stock after deducting $12,000 
for the sinking fund? 

3. The M. & W. Railroad Company with a capital stock of 
$3,750,000, has a mortgage debt of $125,000. Gross earnings for a 
year were $621,000, and operating expenses were $275,000. After 
paying expenses, and 6% interest on the debt, $1000 was added 
to the surplus. The balance of the profits was distributed as divi¬ 
dends. How much should a man receive who owned 25 shares? 

4. A man received $1560 as the annual 12% dividend on stock 
that he owned. He afterward sold 35 shares of the stock at 142J, 
and the remainder of the stock at 143. Making allowance for 
brokerage, and for both the Federal and the New York State 
transfer tax, find the net proceeds of each sale. 

5. A corporation with a capital stock of $100,000 has a net 
profit/ of $16,875.50 for the fiscal year just ended. The directors 
vote to pay a dividend of 7|%, and to put the remainder of the 
profit into the surplus fund, (a) What is the amount of the divi¬ 
dend declared? (6) How much of the profit will be put into the 
surplus fund? (c) What is the amount of dividend that A. L. 
Jones will receive if he owns 230 shares of the stock? 

6. One year ago a man bought 20 shares of a certain stock at 

par ($100), brokerage J%. He has since received 4 quarterly 
dividends each of $1.75 per share. To-day he sells the stock at 
126J, brokerage J%. (a) How much was his total profit? ( b ) 

What rate of interest did he realize on, his investment for the year? 

7. The capital stock of a corporation is $50,000, of which 
$25,000 is 6% preferred stock, and the balance common stock. 
The gross income for 6 months was $11,612, and the expenses were 
$7147. What rate of dividend can be paid on the common stock? 


380 


STOCKS 


8. A man with $37,575 to invest has two propositions to consider. 
They are, (a) shall he deposit his money in savings banks paying 
3|% interest, compounded semiannually, or ( b ) shall he buy 6% 
stock at 166§, brokerage |%? Which is the better proposition, 
and how much better, per annum? 

9. An investor has an option on 2 kinds of stock—one at 89 J, 
paying 5% dividends, and the other at 109J, paying 6% dividends, 
brokerage \% in each case. He decides to buy the 5% stock and 
invests $29,700 in it. How much more or less is his annual 
income than it would have been if he had bought the 6% stock? 

10. A corporation has a capital stock of $7,500,000. The gross 
earnings for a given year were $8,675,921.42, and the expenses 
were $7,562,847.28. To redeem at maturity a bond issue of 
$5,000,000, the company sets aside each year a sum equal to 2\% 
of the face value of the bonds. After setting aside the sinking 
fund and paying the interest at 5% on the bonds, the company 
declares an 8% dividend. The balance of the earnings is put into 
the surplus account. Determine the amount credited to surplus. 

11. A man who owns 480 shares of stock paying regular divi¬ 
dends of 5%, has an offer of 4% stock at 79|. Will it pay him to 
sell the stock he now owns at 99 J, and invest the proceeds of the 
sale in the new stock, brokerage f % each way? If so, how much? 
If not, how much better is the old stock? (No allowance for tax.) 

12. You are offered 525 shares of stock paying regular dividends 
of 5% at a price to yield 6|% on the investment (including broker¬ 
age at |%). What is the market quotation of the stock? How 
much will the 525 shares cost under the conditions set forth? 

13. Mr. Avery sold 175 shares of 7% stock at 168|, brokerage 
$.20 a share, and deposited the proceeds in savings banks paying 
4% interest, compounded semiannually. Make allowance for 
both the Federal and the New York State transfer tax, and 
determine whether his annual income was increased or diminished, 
and how much? 

14. Allowing brokerage as charged on the New York Stock Ex¬ 
change and for the New York State and the Federal transfer tax, 
find the proceeds of a sale of 400 shares of stock at 77f, and 350 
shares at 142 J. 


BONDS 



827. A bond is a secured obligation, promising to pay, for value 
received, a sum of money at a certain time, with interest, payable 
at stated intervals and at a fixed rate. 

828. Bonds are known by the name York 

of the government, municipality, or 
corporation issuing them, by the rate 
of interest they bear, the date of ma¬ 
turity, the purpose for which they were 
issued, and by the type of security 
back of them. For example, U. S. 2% 

Panama Canal loan of 1918-1938, 
means United States bonds paying 
2% interest, due in 1938, payment of 
the bond being optional on any in¬ 
terest date after 1918. City of New 
York 4j’s, 1957, means bonds of the 
City of New York bearing 4J% inter¬ 
est, due in 1957. Chicago, Milwaukee 
& St. Paul General Mortgage 4’s, 1989, 
means bonds bearing 4% interest, 
issued by the Chicago, Milwaukee & 

St. Paul Railway, secured by a general 
mortgage on the property of the com¬ 
pany, and due in 1989. 

Bonds have many names, some of 
which are First Mortgage 5’s, General Mortgage 4’s, Prior Lien 
4£’s, Refunding and Extensions 5’s, etc. If issued by a private 
corporation, they are generally secured by mortgage on the 
property of the company. 

English Consols (consolidated stock) are government bonds of 
England bearing 2|% interest payable quarterly, Jan. 5, April 5, 
July 5, and Oct. 5. These bonds have no definite date of ma- 






382 


BONDS 


829 . 


Illustration of a Bond 






Mm 


■o&c 


•ht+'tb) 


GOLD BOND 


FIRST SHORT 


WILWBM 

under the laws of the Sfotim 
to pay on the first day-cf<fi$ 
the bearer, or if re jpS wB ffii 


ttion organized and existing 
value received, promises 
fitClhe City of New York, to 


gold com of 

est thereon /yOrnlh^irsffia^rTiay^t 

in like gold coin'at the sajd 

annually there after . mtf&fftyuporrpm flj 


MmRgg^^and to pay inter. 
rhur^fQ^fFtmerest to be paid 
•f Nquember, 1913. and semi - 
Pi”. fter'seu* r ally mature. 

which 
law of 


’fd o r require d to pay, or to r etain therefrom u nder any present or /)s 


9 Com • 

made on 
noted on the 

tfferrTW^o bearer■ after which transfer - 
or transferred 
coupons, but the 


pany. 

such books by tS '‘ —■ \ 

bond, but the same may be dist&SVgetf^ 
ability by deliuer^gA HH|HP^HSE| 
to bearer as jft 

same shall be pa 

7 >/j bond~fc3BSggj^BU0Ri 
prodded m said mortf a y^4 
7>/; &W is su&jfK&ff&jBR 
accrued interest as providedfngBttm 
This bond shall not bevomeyBu 
certificate of the said Trustee hereon el 


'Company, and used as 


the face value thereof and 


^shall have been authenticated by the 


3n Z&itness Wbtreof. whlard j. roluns company, has- 

caused these presents to be signed by Its President, and its corporate 





















































BONDS 


383 


turity, but could not be paid before April 5, 1923. They are one 
of the so-called perpetual loans. 

French Rentes are government bonds of France, and bear in¬ 
terest at 3% payable quarterly. These loans are also perpetual loans. 

These bonds are issued in small denominations in order that 
persons with small means may purchase them. 

830. There are two general classes of bonds—coupon and regis¬ 
tered. 

831. Coupon bonds have interest coupons attached. The 
coupons are made payable to bearer, and are dated at regular 
intervals (usually semiannually) and are, in effect, negotiable 
notes promising to pay the interest on the bond as it becomes due. 

832. Registered bonds take their name from the fact that the 
name of the owner is registered with an agent of the maker of the 
bond. On “fully registered” (“registered as to principal and in¬ 
terest”) bonds, interest is paid by check or draft mailed to the 
registered holders. Bonds “registered as to principal only” carry 
interest coupons like an ordinary coupon bond. 



834. The terms, par value, market value, above par, premium, 
below par, and discount, have the same signification as they have 
when applied to stocks. 

835. Stocks and Bonds Compared 

Stocks Bonds 

1 Owner is part owner of corpora- 1. Owner is a creditor of the cor- 

tion. poration. 

2. Owner generally has a vote in 2. Owner generally has no vote, 
the election of directors. 





384 


BONDS 


Stocks 

3. Dividends not due until de¬ 
clared by the directors. 

4. Dividends depend on earning 
power of the corporation. 

5. Purchased, many times, for 
speculation. 


Bonds 

3. Dividends (interest) are pay¬ 
able regularly. 

4. Interest is fixed. 

5. Purchased, generally, for in¬ 
vestment. 


836. Bond quotations are not subject to sudden fluctuations, as 
are stock quotations. The price of bonds depends chiefly upon 
four things: 1st, the security back of the bonds; 2d, the time the 
bond has to run; 3d, interest rates of money; and 4th, the earning 
power of the issuing corporation. 

Notes. Bonds are redeemable at maturity at face value unless otherwise 
specified. 

If interest payments on bonds are not made when due the company is said 
to default in the interest. To default in the bonds is to fail to redeem them 
at maturity. 


837. Following is a partial list of bonds sold on the New York 
Exchange during the week ended July 21, 1923. 


Name of Bond 

Int. Rate 

Date of 
Maturity 

Price 

Am. Tel. & Tel. (Collateral). 

5% 

1946 

971 

Armour & Company . 

41% 

1939 

841 

Bethlehem Steel (Sinking Fund) . . . 

6% 

1948 

981 

Canadian Northern. 

7% 

1940 

1121 

Chi. M. & St. Paul. 

41% 

1989 

81 

Delaware and Hudson. 

51% 

1937 

1081 

General Electric Debentures. 

5% 

1952 

1001 

International Mer. Marine. 

6% 

1941 

801 

Kansas City, Southern . 

5 % 

1950 

851 

Northern States Power (Refunding) 

5% 

1941 

90 

Southern Railway. 

62 % 

1956 

lOlf 

Union Pacific (1st Refunding) .... 

4% 

2008 

841 


The brokerage charge for buying or selling bonds on the New 
York Stock Exchange is as follows: 

For bonds having not over 5 years to run of par value. 
For bonds having over 5 years to run SI.50 per SI000 par value. 
For Government bonds T V%. 





















BONDS 


385 


838. To find the cost of bonds. 

Find the cost of $25,000 Northern States Power 5’s, at price 
quoted on page 384, on July 23, interest dates being April 1 and 
October 1. 

90% of $25,000 = $22,500, cost, flat. 

From April 1 to July 23 = 3 mo. 22 da. 

Interest on $25,000 at 5% for 3 mo. 22 da. = $388.89. 

$22,500 + $388.89 = $22,888.89, cost. 

The cost includes the interest on the face of the bonds since the last preced¬ 
ing interest date. At 90, the bo.nds would cost 90% of $25,000, or $22,500. 
The last interest date was April 1. The interest since that time to July 23 is 
$388.89. The cost is the sum of the “flat” price and the interest, or $22,888.89. 

839. Using the prices quoted on page 384, find the cost, including 
interest, of the following bonds: 



Int. Dates 

Date op 
Purchase 

1. 

6,000 Am. Tel. & Tel. Collateral 5’s . . . 

June—Dec.* 

Oct. 18 

2. 

8,000 Canadian Northern 7’s . 

June—Dec. 

Sept. 20 

3. 

12,000 Delaware & Hudson 5 Ps. 

May—Nov. 

Nov. 15 

4. 

15,000 Chi. Milwaukee & St. Paul 4Ps . . . 

Jan. —July 

Jan. 4 

5., 

25,000 General Electric Debenture 5’s . . . 

Mar.—Sept. 

April 3 

6. 

50,000 Southern Railway 6§’s. 

Apr. —Oct. 

Aug. 25 

7. 

75,000 Union Pacific 1st Refunding 4’s . . 

Mar.—Sept. 

June 10 

8. 

35,000 Kansas City, Southern 5’s. 

Jan. —July 

June 30 

9. 

45,000 Am. Tel. & Tel. Collateral 5’s. . . . 

June—Dec. 

July 18 

10. 

90,000 International Mercantile Marine 6’s 

Apr. —Oct. 

Aug. 11 


840. To find the cost of a bond to yield a given rate. 

At what price will a 5% bond maturing in 3 yr. yield 4%? 
$1025.00 -7- 1.02 = $1004.90, cost if maturity is in 6 mo. 
$1004.90 + $25 = $1029.90, cost plus 6 months’ int. on the cost. 
$1029.90 -T- 1.02 = $1009.70, cost if maturity is in 1 yr. 
$1009.70 + $25 = $1034.70, cost plus 6 months’ int. on the cost. 
$1034.70 ^ 1.02 = $1014.41, cost if maturity is in 1§ yr. 

* Interest is due on the first day of the months named unless otherwise 
specified. 

VAN TUYL’S NEW COMP. AR.—25 
















386 


BONDS 


$1014.41 + $25 = $1039.41, cost plus 6 months’ int. on the cost. 

$1039.41 -f- 1.02 = $1019.03, cost if maturity is in 2 yr. 

$1019.03 + $25 = $1044.03, cost plus 6 months’ int. on the cost. 

$1044.03 -7- 1.02 = $1023.55, cost if maturity is in 2| yr. 

$1023.55 + $25 = $1048.55, cost plus 6 months’ int. on the cost. 

$1048.55 -f- 1.02 = $1027.99, cost if maturity is in 3 yr. 

Hence, the quotation is 102.80. 

If the bond matured in 6 mo., the cost would be a sum which in 6 mo. 
would amount to $1025 (the face of the bond plus 6 months’ interest) at 5% 
interest; that is, the cost is the present worth of $1025 due in 6 mo. at 4%, 
which is $1004.90. If the bond were to mature in 1 yr. instead of 6 mo., 
another semiannual payment of interest would be received. Adding the semi¬ 
annual interest, $25, to $1004.90, gives $1029.90, which is the sum of the cost 
of the bond maturing in 1 yr. plus the interest for 6 mo. at 4% on that cost. 
The cost, then, is the present worth of $1029.90, due in 6 mo. at 4%, 
or $1009.70. 

Continuing the process as shown in the solution, the price to yield 4% in 
3 yr. is shown to be $102.80. 

Note. The above solution is based on the theory that each semiannual 
interest payment of $25 is made up of two separate items, viz., interest on the 
investment at 4% per annum, and a portion of the premium paid for the bond. 
For illustration, take the cost of the bond when maturing in 3 yr. The interest 
for 6 mo. at 4% on $1027.99 equals $20.56. The interest on the bond is $25. 
The difference between $25 and $20.56 equals $4.44. The $4.44 is considered 
as a repayment on account of the premium paid for the bond. Deducting 
$4.44 from $1027.99 leaves $1023.55, which is the cost of the bond if maturing 
in 2£ yr. 

BOND TABLES 

841. Bankers use bond tables to determine the rate of interest 
an investment in bonds will pay, or the price to pay to secure a 
given rate on the investment. The following is a reproduction of 
one page from the “Bond Tables” published by N. W. Harris & 
Co., Bankers, New York and Boston, and is printed with their per¬ 
mission. 

The tables cover periods of time from § yr. to 100 yr., and show 
the net annual rates of interest which bonds paying 7%, 6%, 5%, 
4|%, 4%, 3J%, and 3%, respectively, will pay for 64 different 
quotations on bonds bearing the above rates of interest. 


BOND TABLES 


387 


How to use Bond Tables 

842. l. If a 4% bond matur¬ 
ing in 20 yr. costs 80.64, how 
much will it net the purchaser? 

Find the 4% column in the table 
and follow down the column until 
80.64 is reached. Then follow the 
horizontal line in which 80.64 is 
found to the extreme left-hand col¬ 
umn, and there find 5f in the column 
marked “net per annum.” The 
bond will net 5f % on the cost. 

2. If a man wishes a 4J% 
investment, what price can he 
afford to pay for a 6% bond 
maturing in 20 yr.? 

In the left-hand column find 41, 
and follow the horizontal line in 
which 41 is, to the right until the 
column having 6% at its top is 
reached. The number found in 
both the horizontal line and in the 
vertical column, viz., 119.65 is the 
price he can pay. 

PROBLEMS 

843. Using the table on 
this page, find the price at 
which bonds maturing in 20 yr. can be bought to produce as fol¬ 
lows: 

1. 5% bond to yield 4%. 4. 3|% bond to yield 4§%. 

2. 4i% bond to yield 5%. 5. 5% bond to yield 5J%. 

3. 6% bond to yield 4.6%. 6. 7% bond to yield 5f%. 

Find the rate on the investment on bonds maturing in 20 yr. 
and bought as follows: 

7 . 4J% bonds bought at 96.80. 10. 5% bonds bought at 101.27 
8.4% bonds bought at §8.90. 11. 3|% bonds bought at 71.11. 
9 . 6% bonds bought at 117.82.12. 7% bonds bought at 126.95. 


20 YEARS —Interest Payable Semi-annually. 


BONDS BEARING INTEREST AT THE RATE OF 


An¬ 

num. 

7% 

6% 

5% 

41% 

4% 

31% 

3% 

4 

141.03 

127.36 

113.68 

106.84 

100.00 

93.16 

86.32 

4.10 

* 39-32 

125.76 

112.20 

105.42 

98.64 

91.86 

85.09 

4'/s 

138.90 

126.87 

111.84 

105.07 

98.31 

91.54 

84.78 

4.20 

I 37-63 

124.19 

IIO.75 

104.03 

97 - 3 * 

90.59 

8387 

4* 

136.80 

123.42 

110.04 

103.35 

96.65 

89.96 

83.27 

4-30 

335-98 

122.65 

* 09-33 

102.66 

96.00 

89-34 

82.68 

4 H 

134.76 

121.51 

108.27 

101.65 

95.04 

88.42 

81.80 

4.40 

* 34-35 

121.14 

107-93 

101.32 

94-72 

88.11 

81.51 

4'A 

132.74 

119.65 

106.55 

100.00 

93.45 

86.90 

80.35 

4.60 

131.16 

118.18 

105.19 

98.70 

92.21 

85.72 

79.22 

4 X 

130.77 

117.82 

104.86 

98.38 

91.90 

85.42 

78.94 

4.70 

129.61 

116.74 

103.86 

97-43 

90.99 

84-55 

78.II 

4* 

128.84 

116.02 

103.20 

96.80 

90.39 

83.98 

77.67 

4.80 

128.08 

* 15-32 

*02.55 

96.17 

89.79 

83.40 

•77.02 

v/i 

128.95 

114.27 

101.59 

95.24 

88.90 

82.66 

76.22 

490 

126.58 

113-92 

IOI.27 

94-94 

83 . 6 i 

82.28 

75*95 

6 

125.10 

112.65 

100.00 

93.72 

87.45 

81.17 

74.90 

5 -10 

*23-65 

III.20 

98.76 

92-53 

86.31 

80.09 

73.86 

5/s 

123.29 

110.87 

98.45 

92.24 

86.03 

79‘.82 

73.61 

5-20 

122.22 

109.87 

97-53 

91.36 

85.19 

79.02 

72.85 

5* 

121.51 

109.22 

96.93 

90.78 

84.64 

78.49 

72.34 

5.30 

120.81 

108.57 

96.33 

90.21 

84.09 

77-97 

7 *-85 

6H 

119.77 

107.60 

95.44 

89.36 

83.27 

77.19 

71.11 

5.40 

119.42 

107.28 

95-14 

89.07 

83.01 

76 94 

70.87 

yA 

118.06 

106.02 

93.98 

87.96 

81.94 

75.92 

89.90 

SX 

116.38 

104.47 

92-55 

86.59 

80.64 

74-68 

68.72 


114.74 

102.95 

91.15 

85.26 

79.36 

73.46 

67.57 

sX 

** 3-13 

IOI.46 

89.78 

83-95 

78.II 

72.27 

66.43 

6 

111.56 

100.00 

88.44 

82.66 

76.89 

71.11 

65.33 

6'A 

108.50 

97-*7 

85.84 

80.18 

74-5* 

68.85 

63-*9 

6'A 

105.65 

94.45 

83.34 

77.79 

72.24 

66.69 

61.14 

V/i 

102.72 

9*-83 

80.95 

75-50 

70.06 

64.62 

59-*7 

7 

100.00 

89.32 

78.64 

73.31 

67.97 

62.63 

67.29) 






























388 


BONDS 


13. The amount of the several issues of Liberty Bonds and 
Victory Notes outstanding April 30, 1923, was as follows: 


Name op Loan 

Classification, Interest Rate, 
and Due Date op Issue 

Amount 

1st Liberty Loan 

3f % bonds, 1932-47 

Converted 4% bonds, 1932-47 

Converted 41% bonds, 1932-47 

Second Converted 41% bonds 1932-47 

$1,409,999,050 

10,289,250 

528,022,300 

3,492,150 

2nd Liberty Loan 

4% bonds, 1927-42 

Converted 41% bonds, 1927-42 

44,597,350 

3,223,383,550 

3rd Liberty Loan 

41% bonds, 1928 

3,439,433,500 

4th Liberty Loan 

41% bonds, 1933-38 

6,329,451,850 

Victory Loan 

4f % Notes, 1922-23 

(Mature May 20, 1923) 

768,494,100 


Find the amount required to pay the interest for 1 year. 

14. A man had five $1000 First Liberty Loan 3J% bonds; 
eight $1000 First Liberty Loan Converted 4|% bonds; ten $500 
Victory Loan 4f % notes. What was the value of one coupon on 
each kind of bond or note? What was the total amount of interest 
he collected semiannually? 

12. Interest on the bonds of the Second Liberty Loan is due and 
payable May 15 and Nov. 15. Find the cost of $25,000 of the 
converted 4J% bonds August 13, at 98^f and interest. 

16. A manufacturing company has a capital stock of $150,000, 
and $50,000 of 6% bonds outstanding. After paying the bond 
interest, they declare a dividend of 5%, and still have a surplus of 
$3421.75. What were their gross earnings, if their expenses were 
$21,492.80? 









BONDS 


389 


17. Find the cost, including interest, of $50,000 Canadian 
Northern 7’s on April 3, at 112}, brokerage $1.50 per $1000, 
interest dates being June 1 and Dec. 1. 

18. What investment, including brokerage at $1.50 per $1000, 
in bonds bearing 4|% interest, will be required to produce a gross 
annual interest return of $1800, if the market price of the bonds is 
85f? If, at the end of 4 years, these bonds are sold at 86J, with 
brokerage at $1.50 per $1000, determine the total gross income 
that will have been received from the bonds. (Make no allow¬ 
ance for interest on income.) 

19. A man invested $4475 in municipal 5’s at 89J, no brokerage, 
and at the end of one year, after receiving his interest, he sold 
them at 95J, (no brokerage). What rate per cent did the invest¬ 
ment pay him? 

20. A manufacturing company has outstanding $1,000,000 of 
4J% bonds, and $500,000 of preferred stock entitled to dividends 
at the rate of 6% per annum before anything is paid on the com¬ 
mon stock. The common stock amounts to $2,000,000. The 
net profits for the year are $525,000. What rate of dividend may 
be declared on the common stock if the company sets aside 
$250,000 as a reserve? 

21. Find the cost May 14 of a 6% first Mortgage of $8500 at 
1}% discount, and interest from April 1. 

22. The issued capital stock of a corporation consists of 6000 
shares of common stock and 2000 shares of 6% preferred stock 
($100 par value in each case). The company has also issued 
$150,000 of 5% bonds. The earnings for a given year amount to 
$123,689.15 and the operating expenses to $51,938.40. After pay¬ 
ing 3% of the face value of the bonds for the sinking fund, and pay¬ 
ing the bond interest and the preferred dividend for the year, the 
directors vote the largest whole percent dividend possible on the 
common stock, and put the balance of the earnings into the 
surplus acount. Determine the rate of dividend on the common 
stock and the amount added to surplus. 


EXCHANGE 


844. The system by which merchants in different places dis¬ 
charge their debts to each other without the transmission of money 
is called exchange. 

845. Exchange is of two kinds, viz.: Inland or Domestic, and 
Foreign. 


DOMESTIC EXCHANGE 

846. Domestic exchange treats of the payment of debts between 
merchants in the same country, without the sending of money. 

For instance, if a merchant of New York owes a manufacturer of 
St. Louis for a bill of merchandise, he can pay his debt without 
sending money, in any one of the following ways: 

1. By his personal check. 

2. By a bank draft. 

3. By a commercial draft on one of his debtors. 

4. By a postal money order. 

5. By an express money order. 

6. By a telegraphic money order. 

Note. For the forms of check, bank draft, commercial draft, and note, 
see the chapter on Negotiable Paper, page 314. 

If he sends his check, the manufacturer in St. Louis will deposit 
it in his bank. In due time the check will be returned to the bank 
on which it is drawn, whereupon the amount will be charged 
against the merchant’s account. 

If the merchant sends a bank draft, he must buy and pay cash in 
advance for it. The draft is then sent in the same manner as a 
check. 

Note. Bankers in the City of New York seldom draw drafts on other 
cities. They sell cashier’s or manager’s checks, drawn on their own bank. 

If the merchant draws and sends a draft on one of his debtors, the 
manufacturer must present the draft to the drawee for payment, or 

390 


DOMESTIC EXCHANGE 


391 


he must have his bank collect it for him, and place the proceeds to 
his credit. Or, if it is a time draft, the bank will buy (discount) it, 
after acceptance by the drawee, and credit him at once with the 
proceeds. The bank will then collect the amount of the draft 
from the drawee at maturity, to reimburse itself for the credit given 
the indorser. 

847. New York, Chicago, and San Francisco are the principal 
exchange centers in the United States. 

848. An exchange center is a recognized money center. In it 
are located the correspondent banks of the banks in the surround¬ 
ing cities and towns. 

849. A postal money order is a written order given by the post¬ 
master in one place to the postmaster in another place, to pay a 
specified sum of money to a certain person or to his order. 

850. To obtain a postal money order, one has to make a written 
application stating: 

1. The amount of the order wanted. 

2. The name and address of the payee. 

3. The name and address of the purchaser of the money order. 

851. The largest amount for which a single postal money order 
can be obtained is SI00. The rates charged are as follows: 


$2.50 or less . 

.... 3^ 

$30.01 to $40.00 . . 

. . . 15 i 

2.51 to $5.00 

. . . . 5^ 

40.01 to 50.00 . . . 

. . . 18^ 

5.01 to 10.00 

.... 8^ 

50.01 to 60.00 . . , 

. . . 20^ 

10.01 to 20.00 

. . . . 10^ 

60.01 to 75.00 . . , 

. . . 25 

20.01 to 30.00 

.12 i 

75.01 to 100.00 . . , 

. . . 30 i 


852. More than one indorsement on a postal money order is pro¬ 
hibited by law, except indorsements by banks. 

853. Postal money orders may be presented for payment at the 
post office on which they are drawn, or at any bank. If not pre¬ 
sented for payment within one year from their date, or if lost, the 
Post Office Department at Washington will upon request and 
presentation of evidence of loss, issue a duplicate. 

854. An express money order is a written order of an agent of an 
express company directing another agent of the company to pay a 
certain sum of money to a specified person or to his order. 










392 


EXCHANGE 


855. Express money orders are similar to postal money orders. 
The rates charged for them are the same. Express money orders 
may be indorsed and transferred any number of times, in the same 
manner as checks and drafts. 


Form of Express Money Order 



856. A telepraph money order is an order by telegraph sent by an 
agent of a telegraph company at one place to a telegraph agent at 
another place, directing him to pay a specified sum of money to the 
person named in the telegram, with or without identification, as 
directed by the sender. 

857. The cost of sending a telegraphic money order consists of 
three parts, as follows: 

l. A toll charge, which is the regular charge for a fifteen word 
telegram. See toll charges on next page. 


2. A transfer charge, as follows: 

For $25 or less ..$.25 

Over $25 and not over $50.$.35 

Over $50 and not over $75 $.60 

Over $75 and not over $100.$.85 


After the first $100, up to and including $3000, add $.25 for each 
$100 or fractional part thereof. For amounts above $3000, $.20 
is charged for each $100 or fractional part thereof. 

3. A tax of $.05 on any toll charge of $.50 or less, and a tax of 
$.10 on any toll charge over $.50. 

858. The regular charge for a telegram varies according to 
distance. There is a fixed charge for 10 words or less for each 





































DOMESTIC EXCHANGE 


393 


of the several zones or districts into which the country is divided. 
For each word in excess of 10 words there is an additional charge. 
In the following schedule of rates, the first number is the toll charge 
in cents for 10 words, and the second number is the charge in 
cents for each additional word. 

Toll Charges for Telegrams 


30-2.5 


36-2.5 


42-2.5 


48-3.5 


60-3.5 


72-5 


96-6 


120 - 8.5 


859. Domestic rates of exchange are practically at par through¬ 
out the United States. A charge made by a bank in San Fran¬ 
cisco, for instance, for a draft on New York, is a service or com¬ 
mission charge, and not a premium charge. 

860. To find the cost of a draft or money order. 

1. Find the cost of a draft of $3500 on Chicago if the exchange 
charge is 

of $3500 = $3.50. 

$3500 + $3.50 = $3503.50, cost. 

2. Find the cost of a telegraphic money order of $3875, if the 


toll charge is at the rate of 48 — 3.5. 

The first $100 costs $.85 

The next $2900 costs 29 X $.25 = 7.25 

The next $875 costs 9 X $.20 = 1.80 

The toll charge is $.48 + (5 X $.035) = .66 

The tax is TO 


$10.66 

$3875 + $10.66 = $3885.66, total cost. 

PROELEMS 

861. l. Find the cost of a draft of $2672.50 on New Orleans 
if exchange is at \%. 

2. Find the cost of a telegraphic money order of $1965, if the 
toll charge is at the rate of 60 — 3.5. 














394 


EXCHANGE 


3. A traveling salesman telegraphs for expense money. His 
employer sends him a telegraphic money order for $250. The 
telegraph rate is 30 — 2.5. Find the total cost of the order. 

4. Compare the cost of sending $175 by postal money order, ex¬ 
press money order, and by telegraphic money order, the toll rate 
being 72 — 5. 

5. If the exchange on a draft is §%, which is the cheaper way 
to make payment of $500, by draft, of by telegraphic money order, 
the toll rate being 30— 2.5? 

6. A man pays the following bills: $39.75 by express money 
order; $78.50 by postal money order; $382.50 by telegraphic 
money order (toll rate 42 — 2.5); and $1837.50 by draft, exchange 
\%. Find the total cost of all. 

7. Find the cost of a telegraphic money order of $6475, the toll 
rate being 120 — 8.5. 

862. To find the proceeds of a draft. 

1. Find the proceeds of a sight draft of $2840, collection and 
exchange being |%. 

|% of $2840 = $3.55, collection and exchange. 

$2840 — $3.55 =$2836.45, proceeds. 

Since the bank charges £% for collection, the charge amounts to £% of 
$2840, or $3.55. The proceeds is the difference between $2840 and $3.55, or 
$2836.45. 

2. Find the proceeds of a 60-da. commercial draft of $1500, if 
sold on its date at discount, money being worth 6%. 

$15.00 = the interest for 60 da. 

\% of $1500 = $7.50, discount. 

$15 + $7.50 = $22.50, total discount. 

$1500 - $22.50 = $1477.50, proceeds. 

This solution is identical with that of bank discount. The interest (bank 
discount) is found for the unexpired time of the draft (term of discount). The 
\% discount is equivalent to a collection charge, and is reckoned on the face 
value of the draft. The total discount is the sum of the two discounts, $15 and 
$7.50, or $22.50. The proceeds is the face of the draft less the total discount. 


FOREIGN EXCHANGE 


395 


863. Find the proceeds of the following sight drafts: 



Face 

Collection 
and Exchange 


Face 

Collection 
and Exchange 

1 . 

$ 784.00 

\% 

11. 

$3981.43 

\% 

2. 

1260.00 

i% 

12. 

4289.64 

8% 

3. 

2500.00 

\% 

13. 

1562.50 

\% 

4. 

3675.00 

rV% 

14. 

5283.25 

tV% 

5. 

3875.00 

\% 

15. 

6874.90 

uV% 

6. 

4272.50 

\% 

16. 

9743.85 

ih% 

7. 

1761.40 

1% 

17. 

8498.75 

tV% 

8 . 

2364.82 

\% 

18. 

4964.80 

1% 

9. 

4152.87 

ih% 

19. 

3829.34 

1% 

10. 

7387.95 

\% 

20. 

4382.86 

1% 


21. Find the proceeds of a 90-da. draft of $968.40 sold at \% 
discount, money being worth 6%. 

22. A jobber purchased a bill of goods invoiced at $6483.28, less 
5% for cash. To make immediate payment, he discounted a draft 
of $1864.50 due in 48 da. at |% discount, and 6% interest; a note of 
$2487.25 due in 33 da., at \% discount and 6% interest; and sent 
a bank draft bought at $1.0025 for the balance. How much was 
saved by discounting the invoice and paying cash? 

FOREIGN EXCHANGE 

864. Foreign exchange treats, broadly speaking, of the settle¬ 
ment of international indebtedness by way of bills of exchange 
(checks, etc.), telegraphic transfers, or the actual shipment of gold 
(or silver). It also treats of instruments of credit in the form of 
letters of credit, travelers’ checks, etc. 

865. A bill of exchange is a draft for a stipulated amount drawn 
in one country on another country, payable, according to its tenor 
either on presentation (as checks, demand and sight bills) or after 
a stated period of days or months. It may be a documentary bill 
of exchange; i.e. have documents attached (bills of lading, insur¬ 
ance, inspection, weighing, certificates, invoice, etc.), in evidence 
of the underlying transaction, or it may be a clean bill of exchange; 
i.e. have no documents attached. In the case of a documentary 














396 


EXCHANGE 


draft, the documents, according to a memorandum attached, may 
be delivered to the drawee either on actual payment of the draft 
or on his mere promise to pay the draft at maturity (called accept¬ 
ance), which takes the form of a written notice across the face of 
the draft, “Accepted, payable on (date),” with the signature of 
the acceptor. Before the consignee of the underlying goods can 
get possession of his bills of lading and other documents, he must, 
therefore, either pay or accept—as the case may be—the bill of 
exchange drawn on him. 

866. Bills of exchange are generally drawn in sets of two, called 
either “First and Second of Exchange/’ or in the case of checks 
and demand drafts, “Original and Duplicate.” These are sent by 
different mails, but the payment of one of them cancels the other. 


867. 


A Set of Exchange 






CHEQUE 40191 


■Redmond &Ca. 

Nkw 1-19 



TO PARR'S BANK^mit 

4 BARTHOLOMEW tTAfTK, K. C. 

k LONDON. 


Pm original (duplicate unpaid.) 








ft 


w 

\ CHEQUE 40191 



tRcdmond&dto. 

New ~York .FEB 1-19 

$ 




Ij C£iite4Lu/rvc6iL 


4U4MUU&. 


*%L 


LI CATE (original unpaid) 

\ to PARR’S BANS/Dmited. 

4 Uabthoi.omkw lane, E. c. 

London. 




(MiinU 


wr6 



























FOREIGN EXCHANGE 


397 


Note. Checks on Great Britain having “and Co.” written crosswise on 
their face, are payable only through the mediation of a bank, necessitating 
identification of the payee, and preventing misuse. 

868. Foreign bills of exchange are generally negotiable instru¬ 
ments covering a multitude of international transactions, the most 
important of which are: 

1. The import or export of merchandise from one country into another. 

2. The purchase or sale of securities—public or private—of one country, 
from or to the other. 

3. The investment or loaning of funds in countries (markets) giving a 
higher interest return, and the reflux of such funds upon a change in monetary 
conditions. 

4. The providing of funds or settling of expenses of domestic travelers in 
foreign countries, and the remittances of aliens to their mother countries. 

869. Gold is the universal standard of value in international 
financial transactions, because in each country a fixed value has 
been given to a given weight of gold of a given fineness. 

870. The par of exchange is the value of the pure metal con¬ 
tained in the monetary unit of one country expressed in terms of 
the monetary unit of another country. To illustrate: One pound 
sterling (£1) contains 113.0016 gr. of fine gold. SI contains 23.22 
gr. of fine gold. 113.0016 ^ 23.22 = 4.8665. Hence, £1 = 
S4.8665, called the par of exchange between the United States and 
England. 

871. The rates of exchange fluctuate by certain fixed amounts or 
fractions. Thus, sterling rates vary by $.00 T V 

872. The franc rates generally vary by $.0001. 

873. The par values of the most important exchange rates are 
quoted in the following manner: 


Sterling.1 pound = $4.8665 

French.1 franc = 19.3 £ 

German . . 1 mark = 23.8 i 

Holland.1 guilder = 40.2 i 


874. The principal exchange centers in Europe are London, 
Paris, and Berlin. 






398 


EXCHANGE 


875. A cable transfer is a cabled order to pay a specified sum of 
money to a certain firm or person in a foreign country. In cable 
transfers between private parties, the cost is the cable rate of ex¬ 
change plus |% commission, plus the telegraph and cable charges. 
In cable transfers in trades between bankers, if above a minimum 
amount of $50,000, there is generally no commission or cable charge. 

876. A letter of credit is a circular letter issued by a bank or 
banker, introducing the beneficiary and authorizing him to draw 
money on demand from a large number of institutions all over the 
world up to the inscribed face value of the letter of credit. 

877. A foreign letter of credit is usually issued in sterling, 
this being the currency most readily negotiated the world over. 

878. When wishing to draw funds, the traveler presents his 
letter of credit to the most convenient bank or banker, states the 
amount he desires to draw, and is required to sign a demand draft 
in pounds sterling on the London banker addressed in his credit. 
After careful comparison of his signature with the one appearing in 
his letter of credit, the foreign banker pays over to him the exact 

879. Letter of Credit 


FRONT. REVERSE. 















FOREIGN EXCHANGE 


399 


amount wanted or the equivalent thereof and hands the credit 
back to the traveler. All amounts thus drawn are written off by 
the paying banker on the reverse side of the letter of credit. 

880. A commission of 1% is usually charged by the issuing 
banker in addition to the face value of the letter of credit issued. 

881. Values of Foreign Coins 


Country 

Legal Standard 

Monetary Unit 

Value in 
Terms op U. S. 
Money 

Argentine Republic . . . 

Gold 

Peso 

$0.9648 

Austria-Hungary .... 

Gold 

Krone 

.2026 

Belgium. 

Gold and Silver 

Franc 

.1930 

Brazil. 

Gold 

Milreis 

.5462 

Canada . 

Gold 

Dollar 

1.0000 

Central American States: 




Costa Rica. 

Gold 

Colon 

.4653 

British Honduras . . . 

Gold 

Dollar 

1.0000 

Nicaragua. 

Gold 

Cordoba 

1.0000 

Guatemala 1 




Honduras / 

Silver 

Peso 

.4320 

Salvador. 

Gold 

Colon ' 

.5000 

Chile. 

Gold 

Peso 

.3650 



(Tael 

.7174 to .7991 

China . 

Silver 

\ Dollar 

.5089 to .5204 

Cuba . 

Gold 

Peso 

1.0000 

Denmark. 

Gold 

Krone 

.2680 

France . 

Gold and Silver 

Franc 

.1930 

Germany. 

Gold 

Mark 

.2382 

Great Britain. 

Gold 

Pound Sterling 

4.8665 

Greece . 

Gold and Silver 

Drachma 

.1930 

Italy . 

Gold 

Lira 

.1930 

Japan . 

Gold 

Yen 

.4985 

Mexico . 

Gold 

Peso 

.4985 

Netherlands . 

Gold 

Guilder (Florin) 

.4020 

Norway . 

Gold 

Krone 

.2680 

Panama. 

Gold 

Balboa 

1.0000 

Philippine Islands .... 

Gold 

Peso 

.5000 

Portugal. 

Gold 

Escudo 

1.0805 

Russia. 

Gold 

Ruble 

.5146 

Spain. 

Gold and Silver 

Peseta 

.1930 

Sweden. 

Gold 

Krona 

.2680 

Switzerland. 

Gold 

Franc 

.1930 





































400 


EXCHANGE 


882. A travelers’ check is a circular check which is made payable 
for a stipulated amount in the currency of the foreign countries 
enumerated on the face of the check. Such checks are issued in 
denominations of $10, $20, $50, $100, and $200, and the equiva¬ 
lents in foreign currency are plainly printed on them. 

883. A commission of \% is usually charged by the issuing 
banker or company, with a minimum charge of 30 f. 

884. In the present (1923) unsettled condition of the foreign 
exchange market, it is impossible to treat the subject of exchange in 
a manner adequate to the needs of a time other than the needs 
of the immediate present. At this time German marks are 
quoted at 1,000,000 to $1. Problems involving marks will, 
therefore, be omitted. 

885. Current quotations are given only for demand drafts and 
for cables. So uncertain is the market that 30-, 60-, and 90-day 
quotations are not published. 

886. The following quotations are from the rates issued by The 
Wall Street Journal in July, 1921. 


England, £ 

Par, 4.8665 

Demand 4.58J 
Cables 4.58| 

France, franc 

Par, .193 

Demand 

Cables 

.0589i 

.0590 

Italy, lira 

Par, .193 

Demand 

Cables 

.0436J 

.0437 

Belgium, franc 

Par, .193 

Demand 

Cables 

.0484J 

.0485 

Switzerland, franc 

Par, .193 

Demand 

Cables 

.1781 

.1783 

Holland, guilder 

Par, .402 

Demand 

Cables 

.3940 

.3943 


FOREIGN EXCHANGE 


401 


887. 


Travelers’ Check 



* 000000 ®'* 


AGENCIES 


Ihis Rheque from our 
balance to ihe Order of_ 


| FRANCE f|. 






All the quotations are “direct”; that is, the value of the foreign 
monetary unit is expressed in cents or dollars and cents. Finding 
the cost, therefore, is a simple operation in multiplication. 

888. To find the cost of a bill of exchange. 

1. Find the cost of £450 9s. 6 d. at 4.58. 

6 d. = .5s. 

9.5 s. = £.475 

Therefore, £450.475 

450.475 X $4.58 = $2063.18, cost. 

2. Find the cost of cabling £2500 to London at 4.59|, if the 
commission is i%, and the cablegram charge is for 12 words at 
$.25 a word. 

2500 X $4,595 = $11,487.50, prime cost 

|% of $11,487.50 = $28.72, commission 

12 X $.25 = $3.00, cable charge 

$11,487.50 + $28.72 + $3 = $11,519.22, total cost. 

3. Find the cost of a demand draft of 13,375 lire at the price 
quoted on page 400. 

13,375 X $.04365 = $583.82, cost. 

VAN TUYL’S NEW COMP. AR—26 















402 


EXCHANGE 


PROBLEMS 

889. Using the quotations on page 400, find the cost of: 

1. A demand draft of £875 6s. 8 d. 

2. A demand draft of 16,750 guilders. 

3. A cable for 50,000 Swiss francs, commission |%, and 14 
words at $.30 each. 

4. A demand draft for 35,500 lire. 

5. A manufacturer in Liverpool drew a sight draft on an im¬ 
porter in Philadelphia £859 13s. 8d. How much did it cost the 
importer to pay the draft at 4.59§? 

6. A jeweler imports clocks from Switzerland at a cost of 
F. 14,375. How much must he pay for a demand draft bought 
at $.1781 a franc to settle for them? 

7. Find the total cost of travelers’ checks as follows: 20 checks 
of $10 each; 15 at $20 each; 4 at $50 each; and 3 at $100 each, if 
the issuing bank charges \% commission. 

8. A New York firm owed for imported merchandise as follows: 
From Liverpool £789 5s.; from Paris F. 11,843.50; and from 
Venice 8575 lire. If they bought demand drafts in settlement 
at the rates quoted on page 400, find the total cost. 

9. Haynes & Company of Pittsburg purchased through P. 
Mouquette of Paris a consignment of tea and coffee as follows: 
10 half chests of green tea (600 lb.) at 2 francs a pound; 5 half 
chests of Oolong tea (250 lb.) at 3 francs a pound; 8 mats of Java 
coffee (600 lb.) at 1§ francs a pound. The commission was 2%. 
Find the cost, at the rate quoted on page 400, of a demand draft 
to make full settlement with P. Mouquette. 

890. To find the face of a bill of exchange. 

1. Find the face of demand draft that can be bought for $1000 
at 4.58J. 

$1000 *f■ $4,585 = 218.1025, number of pounds. 

.1025 X 20s. = 2.05s. 

.05 X 12 d. = M. 

Therefore, £218 2s. .6 d. (about 2 far.) 


FOREIGN EXCHANGE 


403 


Since £1 costs $4.58|, $1000 will buy as many pounds as $4.58? is contained 
times in $1000, or 218.1025 times. Reducing the decimal of a pound to 
shillings and pence gives 2s. and 6 d. (about 2 farthings). 

2. How large a demand draft on Florence, Italy, can be bought 
for $1250 at $.044 a lira? 

$1250 -5- $.044 = 28,409.09, number of lire. 

891. Using the rates quoted on page 400, find how large demand 
drafts can be bought for the following amounts: 

1. On London for $1875. 

2. On Paris for $2500. 

3. On Brussels, Belgium, for $2350. 

4. On Amsterdam, Holland, for $4400. 

5. On Berne, Switzerland, for $5000. 

Conversion tables, like the one shown on page 404, are used by 
foreign exchange bankers. These tables reduce to a minimum 
the labor of finding the cost or the proceeds of a bill of exchange. 
A complete table covers many pages. The purpose here is to 
illustrate the use of the table. For that purpose the rate 4.4675 
has been used instead of a rate about par. 

1. Using the table on the next page, find the cost of a draft of 
£407 3s. 6 d. 

In the £400 column and on line 7, find 1818.2725. 

In the 3s. column and the Qd. line find .7818. 

That is £407 costs $1818.2725 

and 3s. 6 d. dosts _ .7818 

Hence, £407 3s. 6 d. costs $1819.0543; that is, $1819.05 

2. How large a draft can be bought for $2252.83 

Find in the table the number of dollars nearest in value to, but less than, 
$2252.83. It is $2251.62, in the £500 column and on line 4, hence £504. 
Next subtract $2251.62 from $2252.83. The difference is $1.21. Now find 
in the lower part of the table the value nearest to $1.21. It is $1.2099, in the 
5s. column and on the 5 d. line. 


Thus: $2252.83 

£504 costs 2251.62 

The difference is 1.21 

5s. 5d. costs 1.2099 


Therefore $2252.83 will buy a draft of £504 5s. 5 d. 




404 


EXCHANGE 


Conversion of Sterling into Dollars and Cents, and vice versa. 


Rate 4.4675 



0 

£100 

£200 

£300 

£400 

£500 

0 


$446.7500 

$893.5000 

$1340.2500 

$1787.0000 

$2233.7500 

1 

$ 4.4675 

451.2175 

897.9675 

1344.7175 

1791.4675 

2238.2175 

2 

8.9350 

455.6850 

902.4350 

1349.1850 

1795.9350 

2242.6850 

3 

13.4025 

460.1525 

906.9025 

1353.6525 

1800.4025 

2247.1525 

4 

17.8700 

464.6200 

911.3700 

1358.1200 

1804.8700 

2251.6200 

5 

22.3375 

469.0875 

915.8375 

1362.5875 

1809.3375 

2256.0875 

6 

26.8050 

473.5550 

920.3050 

1367.0550 

1813.8050 

2260.5550 

7 

31.2725 

478.0225 

924.7725 

1371.5225 

1818.2725 

2265.0225 

8 

35.7400 

482.4900 

929.2400 

1375.9900 

1822.7400 

2269.4900 

9 

40.2075 

486.9575 

933.7075 

1380.4575 

1827.2075 

2273.9575 

10 

44.6750 

491.4250 

938.1750 

1384.9250 

1831.6750 

2278.4250 



0s. 

Is. 

2s. 

3s. 

4s. 

5s. 

0 d. 


$.2234 

$.4467 

$.6701 

$.8935 

$1.1169 

Id. 

$.0186 

.2420 

.4654 

.6887 

.9121 

1.1355 

2d. 

.0372 

.2606 

.4840 

.7074 

.9307 

1.1541 

3d. 

.0558 

.2792 

.5026 

.7260 

.9493 

1.1727 

4 d. 

.0745 

.2979 

.5212 

.7446 

.9680 

1.1913 

5 d. 

.0931 

.3165 

.5398 

.7632 

.9866 

1.2099 

6 d. 

.1117 

.3351 

.5584 

.7818 

1.0052 

1.2286 

7 d. 

.1303 

.3537 

.5770 

.8004 

1.0238 

1.2472 

8 d. 

.1489 

.3723 

.5957 

.8190 

1.0424 

1.2658 

9 d. 

.1675 

.3909 

.6143 

.8377 

1.0610 

1.2844 

10 d. 

.1861 

.4095 

.6329 

.8563 

1.0796 

1.3030 

lid. 

.2048 

.4282 

.6515 

.8749 

1.0983 

1.3216 


i; it 































FOREIGN EXCHANGE 


405 


PROBLEMS 

1. An exporter sold to a broker the following bills of exchange: 
£3472 8s. at 4.5875, and F., 17,425 at $.058f a franc. Find the 
proceeds, if the broker charged \% commission. 

2. A grain exporter in New York drew a bill of exchange on 
a Paris firm for $27,580. If the exchange in Paris on New York is 
at $.0559J a franc, how many francs are required to meet the 
draft? 

3. A wholesaler has drawn on him against an importation of 
woolens a draft of £583 7s. If exchange is at 4.57J, how much does 
it cost him to pay the draft? 

4. A merchant owes bills as follows: £306 4s. Sd.; £410 4s.; 
and £503 3s. lid. Using the conversion table on page 404, find 
the total cost of the three bills. 

5. To pay a bill of $2275, a merchant remits a demand draft in 
pounds sterling bought at the rate used in the table on page 404. 
Find from the table how large a draft he remits. 

6. A Boston firm bought from a firm in Amsterdam 1765 yd. 
of lace at 5§ guilders a yard. A draft in settlement was bought 
at $.3945 per guilder. What was the cost of the draft? 

7. By the death of a relative in Pittsburgh, Pa., a resident of 
Italy becomes entitled to a legacy of $23,450. By his directions 
the money is sent to his agent in London, who remits it to Naples. 
If exchange at Pittsburgh on London is at 4.60, and at London on 
Naples is 104.50 lire to the pound sterling, find how large a draft 
can be sent on to Naples from London. 


UNITED STATES CUSTOMS 

894. Customs, or duties, are taxes levied by the government on 
imported merchandise. 

895. The purpose of the tax is twofold. 

1. For protection of American industries. 

2. For revenue. The income derived from customs is applied toward the 
payment of the nation’s expense. 

896. Customs or duties are of two kinds—specific and ad 
valorem. 

897. “Duties” is the more commonly used term, and will be 
used in this work. 

898. A specific duty is a specified amount levied on each article 
—pound, ton, bushel, gallon, or upon each square foot or square 
yard, etc., as the case may be. 

899. An ad valorem duty is a certain per cent levied on the 
appraised market value of merchandise in the country from which 
it is imported. 

900. Specific duties are levied on some imported articles, ad va¬ 
lorem duties on others, and both are assessed on other kinds of 
merchandise. Specific duties are not reckoned on fractions of a 
unit when the duty on the unit is fifty cents or less. Fractions 
of such units equal to or greater than one half are considered as 
a whole unit. Fractions less than one half are rejected. At some 
ports if the duty on the unit is more than fifty cents, the duty is 
charged on the exact fraction. 

Ad valorem duties are not computed on fractions of a dollar. Fifty cents 
or more is reckoned as a dollar; less than fifty cents is rejected. 

901 . Tare is an allowance made for the weight of the box, bag, 
or other covering material. Leakage is an allowance made for the 
loss of liquids imported in barrels. Breakage is an allowance for 
the loss of liquids imported in bottles. 


406 


UNITED STATES CUSTOMS 


407 


902. The long ton of 2240 pounds is the legal ton in reckoning 
import duties. 

903. A customhouse is a place designated by the government 
where importers of merchandise are to make entry of it and pay 
the duties chargeable thereon, and where vessels are entered and 
cleared. 

904. A vessel is officially entered when the manifest is filed with 
the collector of customs. 

905. A vessel is said to be cleared when the master has received 
his clearance, or permit to leave the port. 

The United States is divided into collection districts. In each district there 
is a customhouse. The port at which the customhouse is located is called a 
port of entry. Vessels and cargoes are entered only at those ports having a 
customhouse. 

A port of delivery is any port at which imported merchandise may be 
delivered after it has been entered at a port of entry. All ports of entry 
are ports of delivery. 

906. At each port of entry is an officer, called the collector of 
customs (popularly called collector of the port), whose duty it is to 
receive entrance papers, issue permits for the clearance of vessels, 
collect the duties, etc. 

907. The public store is a place provided for the examination 
of imported merchandise. 

908. All invoices are made out in the weights, measures, and 
currency of the country from which the merchandise is imported. In¬ 
voices of more than $100 must be certified by a United States consul. 

909. There are two principal forms of entry of merchandise, 
consumption entry and warehouse entry. 

910. A consumption entry is made for all merchandise on which 
the duty is paid at the time of entry. 

911. A warehouse entry is made for all merchandise to be stored 
in a bonded warehouse. 

912. A bonded warehouse is a building for the storage of mer¬ 
chandise on which the duty has not been paid. It derives its name 
from the fact that the owner is required to give a bond to the 
government that he will not deliver from the warehouse any mer¬ 
chandise until the duty has been paid. (With certain exceptions. 
See Art. 915.) 


408 


UNITED STATES CUSTOMS 


913. The importer of the merchandise also gives a bond obligat¬ 
ing himself to pay the duty and to remove the goods within three 
years of the date of entry. Merchandise not removed within the 
three years may be sold for duty and storage charges. 

914. 

IMPORT invoice dated Paris, December 25, 1910 c. c. No. 25,000' 

Entry for Consumption of Merchandise imported by John Doe in the French s.s. Niagara 


whereof ...is master, arrived January 13, 1911 from Havre, new york, January 13, i9ll 


Marks 

Nos. 

PACKAGES AND CONTENTS 
Specifications as per Accompanying Invoices 


35$ 

45$ 



Total 

J.D. 

1/10 

Ten Cases 









Furniture, Mfr. Wood. 

francs 

650.00 







Hardware, Mfr. Metal. 



130.00 


francs 

780.00 





$126.— 

$25.— 



$151.— 





(S 

igned) 

John 

Doe, 








1 Bro 

adway, 

N.Y. 


The above illustration shows the form of entry used when merchandise is im¬ 
ported for immediate consumption. There are ten cases of furniture valued at 
F . 650, or $126, the duty thereon being 35%; and 10 cases of hardware valued 
at F. 130, or $25, the duty on which is 45%. 

915. Merchandise may be withdrawn from a bonded warehouse 
without the payment of the duty for such purposes as: 

1. Exportation. 

2. Transportation and exportation. 

3. Transportation in bond to another warehouse in another district. 

4. Transportation to a manufacturing bonded warehouse. The manufac¬ 
tured product must be exported. 

If merchandise in bond, on which the duty amounting to $50 or 
more has been paid, is exported, the duty less one per cent will be 
returned. The amount so returned is called a drawback. 

916. A tariff is a list of dutiable articles with the legal rate of 
duty on each. 

917. A free list is a schedule of articles on which there is no duty. 

















UNITED STATES CUSTOMS 


409 


918 . Customs regulations provide that the mint par value of 
foreign coins as expressed in dollars shall be used in determining 
the dutiable value of imported merchandise, except that, if such 
value “varies by 5 per centum or more from a value measured by 
the buying rate in the New York market at noon on the day of 
exportation, conversion shall be made at a value measured by 
such buying rate.” 

919 . For values of foreign currency see page 399. 

920 . The following rates of duty are from the Tariff Act of 1923. 

Import Duties 


Articles 

Specific Duty 

Ad Valorem 
Duty 

Brussels Carpet. 


40% 

Butter. 

8 i a pound 


Cheese. 

a pound, but not 


China, decorated. 

less than 25% 

70% 

Cotton clothing, ready made 


35% 

Cotton stockings. 


50% 

Envelopes, plain. 

3 i a pound 

20% 

Hay. 

$4 a ton 


Horses. 

$30, if valued at $150 

20% if value 


or less 

is over $150 

Lead pencils. 

45 ^ a gross 

25% 

Leather gloves, ladies’, not over 



12 inches long. 

$4 a dozen 


Over 12 inches long .... 

$4 a dozen plus 50 i a 



dozen additional for 


Oilcloth. 

each inch in excess 
of 12 inches. But 
in no case less than 
50% nor more than 
70% ad valorem 

3 ^ a sq. yd. 

20% 

Pens, steel. 

12 i a gross 


Plate glass, not exceeding 384 sq. in. 

12 a sq. ft. 


Above 384 sq. in., not exceeding 



720 sq. in. 

15 i a sq. ft. 


All above 720 sq. in. 

17£ a sq. ft. 


Silk ribbons. 


60% 

Woolen clothing, ready made 

30 a pound 

45% 

Writing paper 

3 i a pound 

15% 





















410 


UNITED STATES CUSTOMS 


921. To find a specific duty. 

What is the duty on 250 dozen men's leather gloves? 

250 X $2.50 = $625, specific duty. 

According to the table, page 409, the duty on men’s leather gloves is $2.50 a 
dozen. 250 dozen cost 250 times $2.50/or $625. (Short method, ? of $2500 = 
$625; or 25 2 = 625, hence $625. 


EXERCISES 

922. Using the rates given in the table on page 409, find the 
duty on: 

1. 28,400 lb. of hay at $12 per ton. 

2. 2476 lb. of butter valued at 24 £ a pound. 

3. 400 doz. ladies' leather gloves, 17 inches long at $11 a dozen. 

4. 250 doz. ladies’ leather gloves, 19 inches long at $16 a dozen. 

5. 44 boxes plates glass, each containing 18 pieces 20 in. X 18 
in. 

6. 6 boxes plate glass, each containing 12 pieces 48 in. X 60 in. 

7. 28 gross steel pens. 

8. 800 doz. ladies' leather gloves, 14 inches long at $10 a dozen. 

923. To find an ad valorem duty. 

Find the ad valorem duty on merchandise imported from Eng¬ 
land valued at £560 8s. 10 d., if the duty is 28%. 

£560 8s. = £560.4 
560.4 X $4.8665 = $2727.19 

$2727.19 + $-20 (lOd. = 200 = $2727.39, value in U. S. money. 

28% of $2727 = $763.56, ad valorem duty. 

The value of the invoice must first be reckoned in United States money. It 
amounts to $2727.39. The duty is reckoned on the nearest dollar in the value; 
hence the duty is 28% of $2727, which equals $763.56. 

EXERCISES 

924. Using the rates of duty given on page 409, and assuming 
the value of francs, for duty purposes, at $.08| each, and of 
pounds sterling at par, find the duty on: 

1. Silk ribbons invoiced at F. 2875. 

2. Cotton stockings invoiced at £328 14s. 8 d. 


UNITED STATES CUSTOMS 


411 


3. Decorated china invoiced at F. 12,372. 

4. 1600 sq. yd. of oilcloth invoiced at £50 8s. 

5. Cotton clothing, ready made, invoiced at £649 7s. 6 d. 

6. 800 gross of lead pencils invoiced at £450. 

7. 1800 yd. of brussels carpet invoiced at F. 9 a yard. 

8. 1400 lb. of cheese invoiced at F. 2750. 

9. 10 horses valued at $175 each. 

10. 1500 lb. of writing paper invoiced at F. 5 a pound. 

11. Ready made woolen clothing weighing 625 lb. and invoiced 
at £387 12s. 

12. 2500 lb. of plain envelopes invoiced at F. 10,500. 

PROBLEMS 

925. 1. A merchant imported merchandise from Paris amount¬ 
ing to F. 13,878. The duty was 27J%. If francs, for duty pur¬ 
poses were valued at $.08f, and a bill of exchange in settlement 
was bought at $.08f, find the total cost. (See Art. 918.) 

2. A stationer imported 1000 reams of writing paper, weighing 
\\ lb. to the ream at F. 7.5 a ream. If francs were valued at 
$.09§, for duty purposes, and sight exchange was selling at 
$.08|, find the cost of a sight bill, to settle for the paper, and the 
total cost. 

3. An importer purchased 300 gross of lead pencils in London 
at 16s. 6c?. a gross. At what price per gross should he sell them to 
gain 40% of total cost, if he paid for the pencils with a bill of 
exchange bought at 4.7320? 

4. An invoice of Brussels carpet contains 3480 yd. at 7s. 6 d. 
a yard. Transportation and other charges except duty amount 
to $125. At what price a yard should the carpet be sold to net a 
profit of 33J% of the sales, if a bill of exchange in settlement 
was bought at 4.7130? 

5. An importation of cotton stockings valued at £712 14s. 
contains 2250 dozen pairs. At what price per pair should they 
be sold to gain 37£% of the selling price, if exchange in settlement 
was bought at 4.6730? 


EQUATION OF ACCOUNTS 


926. If you buy a bill of merchandise valued at $1200 on a credit 
of 60 days from Aug. 1, when is the bill due? If you pay the face 
value of the bill Aug. 31, how many days* discount could you 
claim? If you do not pay the bill till Oct. 30, how many days’ 
interest could the seller claim? 

If you pay $600 on account Aug. 31, how long after the maturity 
ought you to be allowed to retain the other $600 of the bill, to bal¬ 
ance your loss by paying $600 Aug. 31? 

927. A merchant bought goods Apr. 10 worth $400, terms cash; 
and $400, terms 20 days. On what date may both bills be paid in 
one amount without loss of interest to either party? 

928. Equation of accounts is the process of finding the date 
when (1) several items due at different dates may equitably be 
paid in one amount, or (2) the balance of an account may_ be 
equitably settled. 

929. Accounts are generally divided into two classes, viz.: those 
having items on but one side, the equating of which is called simple 
equation, or equation of payments, and those having both debit 
and credit items, the equating of which is called compound 

equation, or equation of accounts. 


EQUATION OF PAYMENTS 

930. The theory on which equation of payments is based is that 
the interest on overdue items in an account is canceled by the dis¬ 
count on other items in the account paid before they are due. 

To illustrate the theory, suppose a man owes $200 due May 10, 
and $600 due May 30. By equation, the entire amount may be 
paid May 25, because the interest on $200 from May 10 to May 25 
is equal to the discount on $600 from May 25 to May 30. 


412 


EQUATION OF PAYMENTS 


413 


931. To find the equated date of a one-sided account. 

The following example will illustrate several terms and their 
definitions: 

Find the equated date of the following bills: 

July 8, S600; July 20, $1200; Aug. 17, 1800. 


July 

8, 

$ 600 

40 da. 

$4.00, interest. 

July 

20, 

$1200 

28 da. 

$5.60, interest. 

*Aug. 

17, 

$1800 

0 da. 

$0.00, interest. 



$3600 


$9.60, interest. 


Interest on $3600 for 1 da. = $3,600 -5- 6 = $.60. 

$9.60 -7- $.60 = 16 da., average term of credit. 

Aug. 17 — 16 da. = Aug. 1, equated date. 

Arrange the several items and their respective due dates as shown in the 
solution. For the purpose of the solution, the latest date, Aug. 17, is assumed 
as the date of settlement. The assumed date of settlement is called the focal 
date .* If the $600 due on July 8 is not paid till Aug. 17, it is overdue 40 da., 
and the accrued interest amounts to $4. So, also, if the $1200 due July 20 is 
not paid till Aug. 17, it is overdue 28 da., and the interest is $5.60. If the 
$1800 is paid on its due date, Aug. 17, there is no interest charged or discount 
allowed. Now, if the total amount of the bill, $3600, is paid on Aug. 17, the 
seller loses $9.60 interest on the overdue items. To avoid this loss the account 
should have been settled earlier. 

The interest on $3600 for 1 da. is $.60; hence, if the payment had been made 
one day earlier, the loss to the seller would have been $.60 less. Therefore, the 
account should have been settled as many days before Aug. 17, as $.60, the 
interest on the debt for 1 da., is contained times in $9.60, the amount of interest 
due, which is 16 times, or 16 da. Counting back 16 da. from Aug. 17, gives 
Aug. 1, the equated date A 

To prove the correctness of the work and to show there is neither loss nor 
gain to either party, find the interest on all items due before Aug. 1 from the 
date they are due to Aug. 1. Then find the discount (simple interest) on the 
item due after Aug. 1 from Aug. 1, to its due date. Thus: 

* Any date may be used as a focal date. In practice either the earliest or 
latest date is generally used, owing to the fact that they are the most con¬ 
venient dates. In this book, the latest date is used. 

t The equated date, the average date of payment, the average due date, 
and similar expressions are synonymous, and are the names given the date of 
settlement as determined by equation. 











414 


EQUATION OF ACCOUNTS 


The interest on $600 from July 8 to Aug. 1, 24 da., is $2.40 
and the interest on $1200 from July 20 to Aug. 1, 12 da., is $2.40 

Total interest due $4.80 

The discount on $1800, from Aug. 1 to Aug. 17, 16 da., is $4.80; that is, the 
interest on the overdue items is equal to the discount on the item that was pre¬ 
paid. Hence, the work is correct, and Aug. 1 is the average due date. 


PROBLEMS 


932. Find the average due date of the following bills: 


1924 

1. Mar. 1, $300 
Mar. 10, $600 
Mar. 20, $900 

1924 

4. May 1, $1500 
May 20, $600 
June 8, $300 
June 20, $900 


1924 

2. Sept. 15, $800 
Sept. 30, $900 
Oct. 15, $300 

1924 

5. June 18, $560 
June 30, $720 
July 15, $900 
July 25, $1000 


1924 

3. Dec. 1, $1200 
Dec. 8, $1500 
Dec. 24, $3000 

1924 

6. Aug. 10, $360 
Aug. 20, $480 
Aug. 28, $660 
Sept. 15, $840 


7. On what date may the following items be paid in one amount 
without loss of interest to either party? 


Jan. 3, 1924, Mdse., 30 da., $600 
Jan. 14, 1924, Mdse., 1 mo., $900 
Jan. 20, 1924, Mdse., 60 da., $800 
Feb. 10, 1924, Mdse., 1 mo., $300 


1924 1924 

Jan. 3 + 30 da. = Feb. 2, 

$600 

47 da. 

$4.70 

Jan. 14 + 1 mo. = Feb. 14, 

$900 

35 da. 

$5.25 

Jan. 20 + 60 da. = *Mar. 21, 

$800 

0 da. 

$.00 

Feb. 10 + 1 mo. = Mar. 10, 

$300 

11 da. 

$.55 


$2600 


$10.50 


The interest on $2600 for 1 da. = $2,600 -f- 6 = $.4333. 

$10.50 -f- $.433 = 24.2; 24 da., t average term of credit. 

Mar. 21 — 24 da. = Feb. 25, equated date. 

When various terms of credit are given on the items to be equated, it is neces¬ 
sary first to find the due date of the separate items. Having found the several 
due dates, we perform the solution as in the preceding exercises. 

t A fraction of a day less than \ is rejected; a fraction greater than \ is 
called 1 da 















EQUATION OF ACCOUNTS 


415 


Find when the following bills may be equitably paid in one 
amount. Prove the work. 


1924 8 

Apr. 1, Mdse., 15 da., $500 
Apr. 10, Mdse., 1 mo., $800 
May 1, Mdse., 30 da., $1000 
May 10, Mdse., Net, $300 


1924 9 

July 1, Mdse., 1 mo., $856.60 
July 18, Mdse., 2 mo., $1260.00 
July 30, Mdse., 30 da., $1000.00 
Aug. 10, Mdse., 10 da., $1268.40 


10. Daniel Reardon bought of Lord & Taylor merchandise on 60 
days’ credit as follows: Aug. 12, 1924, a bill for $1388.75; Sept. 10, 
a bill for $1564.25; and Sept. 20, a bill for $975.50. Reardon 
wishes to give a 2-months’ interest-bearing note! in settlement of 
the entire amount. When should the note be dated? Write the 
note. 

EQUATION OF ACCOUNTS 


933. To find the equated date of a two-sided account. 


l. What is the equated date of payment of the balance of the 
following account? 

1924 1924 

May 1, Mdse., $1840 May 10, Cash, $600 

June 1, Mdse., $1200 May 20, Cash, $900 


Due 

Amt. 

Days 

Int. 

Due 

Amt. 

Days 

Int. 

1924 
May 1 

$1840 

31 

$9,506 

1924 
May 10 

$600 

22 

$2.20 

tJune 1 

$1200 

0 

0.000 

May 20 

$900 

12 

$1.80 


$3040 


$9,506 


$1500 


$4.00 


$3040 — $1500 = $1540, balance due on the account. 
$9,506 — $4.00 = $5,506, balance of interest due. 

Interest on $1540 for 1 da. = $1,540 -J- 6 = $.2566. 

$5,506 -4- $.2566 = 21.4, no. da., 21 da. 

21 da. back from June 1 = May 11, the equated date. 

All the debit items and all the credit items are arranged as shown in the solu¬ 
tion. The latest date on either side is taken as a focal date. If the bill of $1840 
were not paid until June 1, it would have been 31 da. overdue, and there would 

t An account becomes interest bearing at the equated date. 























416 


EQUATION OF ACCOUNTS 


have been $9,506 interest due on it. There would be no interest due on the bill 
of $1200 if paid on June 1. The total interest due, then, by settling the account 
on June 1, would be $9,506. But two payments have been made. The $600 
paid on May 10 was paid 22 da. before the assumed date of settlement, hence, 
the total amount of interest due as shown on the debit side is reduced by the in¬ 
terest on $600 for 22 da., or $2.20. In like manner the $900 paid on May 20, 
or 12 da. before June 1, reduces the debit interest by $1.80 more. The total 
credit interest is $4, leaving a balance of $5,506 interest due. The balance of 
the account is $1540. The interest on $1540 for 1 da. is $.2566, hence if the 
assumed date of settlement had been 1 da. earlier, the balance of interest would 
have been $.2566 less. To find the number of days earlier than June 1, which 
the account should have been settled, divide $5,506 by $.2566, which gives 21 
da. Therefore the equated date is 21 da. before June 1, or May 11. 


Proof 


Due 

Amt. 

Days 

Int. 

Dis. 

Due 

Amt. 

Days 

Int. 

Dis. 

1924 
May 1, 
June 1, 

$1840 

$1200 

10 

21 (Dis.) 

$3,066 

$4.20 

1924 
May 10, 
May 20, 
May 11, 

$600 

$900 

$1540 

1 

9 (Dis.) 
0 

$.10 

.00 

$1.35 

$3040 

$3,066 

$4.20 

$3040 

$.10 

$1.35 


$4.20 — $3,066 = $1,134, debit discount. 

$1.35 — $ .10 = $1.25, credit discount. 

$1.25 — $1,134 = $.116, less than half a day’s interest on the 
balance, $1540. 

Write the balance, $1540, with equated date on the credit side of the account 
as shown. Equating the account with May 11 as the focal date, and balancing 
the items of interest and discount, makes the credit side of the account $.116 
more than the debit side. Since the interest on $1540 for 1 da. is $.2566, the 
difference between the two sides is less than half the interest for 1 da. Inas¬ 
much as the law does not recognize a fraction of a day, the account as it now 
stands is considered as being in balance, and the solution proves. 

2. Equate the following account: 


Charles R. Goodrich 


1924 





1924 





June 

10 

Mdse. 60 da. 

800 


June 

30 

Cash 

1000 



25 

Mdse. 3 mo. 

1260 


July 

10 

Note, 2 mo. 

1000 


July 

1 

Mdse. 1 mo. 

900 







































EQUATION OF ACCOUNTS 


417 


Due 

Amt. 

Days 

Int. 

Due 

Amt. 

Days 

Int. 

1924 




1924 




Aug. 9, 

$800 

47 

$6,267 

June 30, 

$1000 

87 

$14.50 

i Sept. 25, 

$1260 

0 

0.00 

Sept. 10, 

$1000 

15 

$2.50 

Aug. 1, 

$900 

55 

$8.25 






$2960 


$14,517 


$2000 


$17.00 


$2960 — $2000 = $960, balance of the account. 

$17.00 — $14,517 = $2,483, credit balance of interest. 

Interest on $960 for 1 da. = $.960 -s- 6 = $.16. 

$2,483 ^ $.16 = 15.5; 16 da. 

Sept. 25 + 16 da. = Oct. 11, equated date. 

First find the due dates of the several items having terms of credit. 60 da. 
after June 10 is Aug. 9; 3 mo. after June 25 is Sept. 25; and 1 mo. after July 1 
is Aug. 1. The several amounts due are written opposite their respective due 
dates. On the credit side of the account, the cash was paid June 30, but the 
note is not due until Sept. 10. 

The latest date on either side is taken as a focal date, which is Sept. 25. If 
the account were not settled until Sept. 25, the $800 due Aug. 9 would be 
overdue 47 da., the interest on which would be $6,267. So also, the $900 due 
Aug. 1 would be 55 da. past due, with accrued interest of $8.25. Against these 
charges of interest there stand on the credit side $1000 paid June 30, or 87 da. 
before the assumed date of settlement, the interest on which is $14.50, and the 
note of $1000 due and paid Sept. 10, 15 da. before Sept. 25, the interest on 
which is $2.50. 

By balancing the account it is found that while there is $960 due the holder 
of the account, he is owing a balance of $2,483 interest on payments made 
before the assumed date of settlement. Hence, the debtor has a right to keep 
the balance of the account, $960, as many days after Sept. 25 as it will take 
$960 to earn the balance of interest, $2,483. $960 earns $.16 a day and to 

earn $2,483 will require 16 da., which added to Sept. 25 gives Oct. 11 as the 
equated date. 

The proof is the same as in the preceding example. 

934. Counting back or forward from the focal date. 

l. When to count back. 

If the balance of the account and the balance of interest are on 
the same side of the account, that is, both on the debit side, or both 
on the credit side, the person who owes the balance of the account 
owes the balance of interest also; hence he should have settled the 

account earlier. 

VAN TUYL’S NEW COMP. AR.—27 
























418 


EQUATION OF ACCOUNTS 


2. When to count forward. 

If the balance of the account and the balance of interest are on 
opposite sides of the account, that is, one balance on the credit 
side, and the other on the debit side, the person who owes the bal¬ 
ance of the account has the balance of interest due him; hence he 
has the right to keep the balance of the account after the focal date 
long enough to earn the balance of interest. 

935. Equate the following accounts: 

1. C. H. Sturgis 


1924 





1924 




Sept. 

3 

Mdse. 

720 


Sept. 

6 

Cash 

600 


18 

U 

960 



25 

U j 

1000 

Oct. 

20 

it 

600 







2. D. H. Oglethorpe & Son 






1924 




2 

Mdse. 

1150 


Jan. 

10 

Cash 

500 

24 

“ 

900 



15 

U 

600 

8 

u 

780 


Feb. 

1 

U 

1000 

16 

K 

1500 







3. L. D. Thornburg & Co. 


1924 





1924 




Nov. 

1 

Mdse., 60 da. 

840 


Dec. 

7 

Cash 

1500 


15 

“ 30 da. 

560 


1925 




Dec. 

1 

“ 3 mo. 

1860 


Jan. 

4 

U 

1000 


15 

“ 1 mo. 

1000 



20 

Check 

500 


4. 


D. L. Sutherland 


1924 

Feb. 

1 

Mdse., net 

125 


11 

“ 30 da. 

450 


20 

“ 60 da. 

750 


28 

“ 30 da. 

1000 


1924 




Feb. 

8 

Cash 

100 

Feb. 

15 

Note, 30 da. 

500 

Mar. 

1 

Note, 60 da.J 
with int. 

1200 




























































CASH BALANCE 


419 


CASH BALANCE 

936. Cash balance, or accounts current, treats of finding the 
balance due on an account on a given date. 

937. Interest accrues, and is collectible, on overdue accounts. 
The debtor is entitled to discount on all items paid before maturity. 

Retailers seldom charge interest on overdue accounts except on 
balances brought down quarterly, semiannually, or annually, and 
then by agreement. Wholesalers generally do charge interest on 
overdue accounts. 

938. If an account has been equated, the balance due on a given 
date is the balance of the account plus the interest or less the dis¬ 
count for the time between the equated date and the date of settle¬ 
ment, according as the date of settlement is after or before the focal 
date. Generally, however, the cash balance of an account is de¬ 
termined as shown in the following example: 

939. If money is worth 6%, find the cash balance of the follow¬ 
ing account Jan. 1, 1925. 


John R. Caldwell 


1924 





1924 





July 

8 

Mdse., 2 mo. 

1260 


Sept. 

15 

Cash, 

1000 


Sept. 

1 

“ 3 mo. 

1848 

75 

Oct. 

10 

Cash, 

1200 


Nov. 

10 

“ 2 mo. 

600 


Dec. 

1 

Note, 10 da. 

500 




Amt. 

Days 

I NT. 

Dis. 


Amt. 

Days 

Int. 

1924 





1924 




Sept. 8 

$1260 

115 

$24.15 


Sept. 15 

$1000 

108 

$18.00 

Dec. 1 

$1848.75 

31 

$9.55 


Oct. 10 

$1200 

83 

$16.60 

1925 





Dec. 11 

$500 

21 

$1.75 

Jan. 10 

$600 

9 dis. 


$.90 






$3708.75 


$33.70 

$.90 


$2700 


$36.35 


$3708.75 + $33.70 - $.90 = $3741.55, total debit. 
$2700 + $36.35 = $2736.35, total credit. 

$3741.55 - $2736.35 = $1005.20, balance due. 






































420 


EQUATION OF ACCOUNTS 


Write all items, both debit and credit, with their respective due dates as 
shown in the solution. Find the number of days from the due date of each 
item to Jan. 1,1925. Thus, the time from Sept. 8 to Jan. 1 is 115 da., and the 
interest on $1260 for 115 da. is $24.15. The $1848.75, due Dec. 1, is overdue 
31 da., and $9.55 interest has accrued. The $600 due Jan. 10 is due 9 da. after 
Jan. 1, and the debtor is entitled to $.90 discount if it is paid Jan. 1. On 
the credit side, $1000 was paid 108 da. before the day of settlement, and in 
that time has earned $18 interest. The $1200 was paid 83 da. before settle¬ 
ment, and has earned $16.60 interest, and the $500 has earned $1.75 interest. 

The gross amount due, including interest and discount, is $3741.55, against 
which there has been paid $2700 plus the total credit interest, $36.35, making 
$2736.35. The balance of the account is, therefore, $3741.55—$2736.35 or 
$1005.20. 

Note. Compare this solution with that of partial payments by the 
Merchants’ Rule, page 341. 


PROBLEMS 


940. l. How much is due on the following account July 1, 1924, 
interest at 6%? 

L. M. Boardman 


1924 





1924 





Jan. 

1 

Balance 

1328 

75 

Feb. 

1 

Cash, 

1000 


Mar. 

10 

Mdse., 3 mo. 

1475 


May 

10 

U 

1200 


Apr. 

1 

“ 30 da. 

875 


June 

10 

U 

800 



2. Find balance due Oct. 1, 1924, at 5%. 


S. L. Putzman 


1924 

May 

1 

Balance, 

2896 

50 

1924 

Aug. 

14 

Cash, 

2500 


June 

1 

Mdse., 2 mo. 

1824 

70 

June 

20 

U 

1000 


July 

12 

“ 10 da. 

1022 

60 

Sept. 

1 

(( 

1000 



3. How much is due Apr. 1, 1924, interest at 7%? 


M. N. Orson 


1924 

Jan. 

18 

Mdse., 2 mo. 

4827 

50 

1924 

Feb. 

20 

Cash, 

4000 

Feb. 

12 

“ 1 mo. 

3600 


Feb. 

20 

Note, 2 mo. 

3000 

Mar. 

1 

“ 10 da. 

1200 


















































PARTNERSHIP 


941. A partnership is an association of two or more persons for 
the purpose of carrying on a legal business and dividing the profits. 

942. The profits, or losses, of a partnership are shared according 
to an agreement made in advance. The agreement may be to 
share equally, according to investment, according to average in¬ 
vestment, or according to some other specified proportion. 

943. A general partner is one who takes an active part in the 
business and who is known to the public as a partner. 

944. A nominal partner is one who is held forth as a partner, and 
as such is liable to innocent third parties for the liabilities of the 
firm. 

945. A dormant, or secret partner is a general partner with the 
exception that his connection with the partnership is unknown to 
the public. If his membership in the firm becomes known, he is 
liable for the debts of the partnership. 

946. A special or limited partner is one whose liability for the 
debts of the firm is limited to the amount of his investment. 

947. The capital of a firm consists of all money or other form of 
property invested in the business. 

948. The resources of a firm consist of all money and property 
belonging to the firm, and all debts or obligations owing to the 
firm. 

949. The liabilities of a firm consist of all its debts or obligations 
to others. 

950. The present worth of a firm is the excess of its resources 
over all its liabilities. It is also the sum of the net investment 
and the net gain. 

951 . The net investment of a firm is the difference between the 
total investments and the total withdrawals. 


421 


422 


PARTNERSHIP 


952. The insolvency of a firm is the excess of its liabilities over 
its resources. 

953. The average investment of a firm is a sum which, if placed 
at interest for a given unit of time (day, month, or year) will pro¬ 
duce the same amount of interest as the various amounts invested 
by the partners for different lengths of time. 

954. The net gain is the excess of gains over losses. 

955. The net loss is the excess of losses over gains. 

956 Apportionment of gains and losses when investments are 
made for equal periods. 

EXAMPLE 

1. A and B each invest $5000. They gain $3000 in one year. 
Find gain of each. 

$5000 + $5000 = $10,000, total investment. 
i 5 o °o 0 o°o = h each partner’s share of the gains. 

^ of $3000 = $1500, each partner’s gain. 

PROBLEMS 

957. 1. Three partners invest $3000, $4000, and $5000, re¬ 
spectively, for two years. Their gain is $15,000. Find the 
present worth of each at closing. 

2. C and D begin business with a joint capital of $12,000. At 
the end of 4 years their present worth is $28,000. Find each part¬ 
ner’s gain if they share equally. 

3 . Jan. 1, 1924, a firm’s liabilities exceeded its resources by 
$1200. Two years later its present worth was $15,000. If there 
were three partners sharing equally, with what amount of gain 
should each be credited? 

4 . Two men began business together with $3000 borrowed 
money. In three years’ time their resources exceeded their liabili¬ 
ties by $12,000. What was their average annual gain? 

5 . Three partners’ interests in a business were in the ratio of 4, 
5, and 6. During a certain year they gained $13,500. Their 
present worth at the end of the year was $9000. What was the 
condition of each partner’s account at the beginning of the year? 

6. Two partners, whose interests were in the ratio of 2 and 3 
suffered a loss of $8000 in the year 1924. Dec. 31, 1924, the firm 



PARTNERSHIP 


423 


was insolvent $3000. What was the condition of each partner’s 
account Jan. 1,1924? 

7 . E and F share equally in gains and losses. E has invested 
$6000, and F $7000. At the end of one year their resources amount 
to $24,000, and their liabilities to $5000. Find each partner’s 
present worth. 

8. The net gain of a firm for one year is 25% of the original 
investment. At the close of the year the present worth of the firm 
is $25,000. If there are two equal partners, find the net gain of 
each. 

9. Messrs. Brown & Brooks form a partnership with an invest¬ 
ment of $16,000, one half of which is cpntributed by each partner. 
Their net gains and net losses for three years are as follows: first 
year, net gain $6000; second year, net loss $2000; third year, net 
loss $10,000. Find the condition of each partner’s account at the 
close of each year. 

10. Three partners, A, B, and C, invest $8000, $9000, and $10,000, 
respectively. They are to share the gains and losses in proportion 
to their investments. Their total gains for a year are $16,000; 
their losses are $5200. Find each partner’s present worth at the 
end of the year. 

11. X, Y, and Z, are partners sharing gains and losses according 
to investment. X invests $5000; Y $8000; and Z $7000. At the 
end of the first year of business the firm’s present worth is $32,000. 
Find each partner’s gain. 

12. D, E, and F are partners. D is to receive f of the gains, 
E and F£. Their total investment is $16,000. At dissolution 
they have $14,800 in the bank, $8000 worth of merchandise, and 
notes on hand $2200. How shall the net gain be divided? 

13 . Parker, Rogers, and Robinson share gains and losses accord¬ 
ing to investment. Parker invests $7000, Rogers $8000, and 
Robinson $9000. At the close of the first year, their ledger shows 
gains and losses as follows: merchandise, gain $13,600; expense, 
loss $4700; interest, gain $221.60; freight, loss $87.50; real 
estate, gain $1234.30. What is each partner’s present worth? 


424 


PARTNERSHIP 


958. Apportionment of gains and losses according to average 
investment. 

A and B enter into partnership agreeing to share gains and 
losses according to average investment. A invests, Jan. 1 , 1924, 
$4000; Apr. 1, $3000; and withdraws, Sept. 1, $2000. B invests, 
Jan. 1,1924, $5000; June 1, $4000; and withdraws, Oct* 1 , $2000. 
Dec. 31,1924, they have a net gain of $8940. Find each partner’s 
present worth. 

A 


1924 

Sept. 

Dec. 

1 

31 

Present Worth 

2000 

9020 


1924 

Jan. 

Apr. 

Dec. 

1 

1 

31 

Net Gain 

Present Worth 

4000 

3000 

4020 


11,020 


11,020 






1925 

Jan. 

1 

9020 


B 

1924 





1924 





Oct. 

1 


2000 


Jan. 

1 


5000 


Dec. 

31 

Present Worth 

11920 


June 

1 


4000 







Dec. 

31 

Net Gain 

4920 





13,920 





13,920 







1925 










Jan. 

1 

Present Worth 

11,920 



A invests $4000 for 3 mo. = $12,000 for 1 mo. 

7000 for 5 mo. = 35,000 for 1 mo. 

5000 for 4 mo. = 20,000 for 1 mo. 

which = $67,000 for 1 mo. 

B invests $5000 for 5 mo. = $25,000 for 1 mo. 

9000 for 4 mo. = 36,000 for 1 mo. 

7000 for 3 mo. = 21,000 for 1 mo. 

which = $82,000 for 1 mo. 

$67,000 + $82,000 = $149,000, firm’s total investment for 1 mo. 
A’s share of the gain = ^ of $8940, or $4020. 

B’s share of the gain ^9 of $8940, or $4920. 

A’s investment, Jan. 1, 1924, of $4000 remains unchanged until Apr. 1, a 
period of 3 mo., which is equivalent to an investment of $12,000 for 1 mo. 






















































PARTNERSHIP 


425 


Apr. 1, he invests $3000 more, making his total investment $7000, which 
remains unchanged until Sept. 1, or 5 mo., and is equivalent to an investment 
of $35,000 for 1 mo. Sept. 1, he withdraws $2000, leaving a balance of $5000 
invested in the business for the remainder of the year, or 4 mo., and equivalent 
to $20,000 for 1 mo. A’s average investment, then, is equivalent to an invest¬ 
ment of $67,000 for 1 mo. 

In the ' me way B’s average investment is found to be equivalent to an in¬ 
vestment of $82,000 for 1 mo. Hence, the firm’s average investment for the 
year is equivalent to $149,000 invested for 1 mo. Therefore, 

A’s share of the gain is T V^ of $8940, or $4020, and B’s share of the gain is 
of $8940, or $4920. 

The simplest method of determining the condition of the partner’s accounts 
is to write them in the form of ledger accounts, as shown in the solution. Each 
partner is credited with his investments and charged with his withdrawals. He 
is then credited with his net gain (or charged with his net loss) and the account 
is closed the same as in a ledger. 

PROBLEMS 

959. l. Three men are in partnership and make a profit of 
$16,350. A invests, Jan. 1, 1924, $3000, and July 1, $2000. B 
invests, Jan. 1, 1924, $4000, May 1, $2000, and Sept. 1, $1000. 
C invests, Jan. 1, 1924, $8000, Mar. 1, $3000, and withdraws, 
Aug. 1, $4000. Find the present worth of each partner, Jan. 1,1925. 

2. C and D are partners; C invests $1200 for 5 mo., and then 
increases his investment to $2000 for 5 mo. D invests $1500 for 
6 mo., and then withdraws $500 for 4 mo. At the expiration 
of 10 mo. they dissolve partnership, having resources amount¬ 
ing to $8800. To how much of the resources is each partner 
entitled? 

3. E and F begin business as partners, Jan. 1,1923, each invest¬ 
ing $10,000. Jan. 1, 1924, G is admitted into the firm with an 
investment of $12,000. E increases his investment $5000, and F 
invests $8000 more on the day G is taken into the business. Jan. 1, 
1925, they have resources amounting to $78,000 and liabilities, 
$7000. Find the condition of each partner’s account. 

4. A and B engage to perform a given piece of work for $580, the 
work to be completed in 40 da. A has furnished 3 men for 22 
da., and B 4 men for 19 da. Only 10 da. of time remain, and in 
order to finish the work on time, A engages 5 extra men, and B 4 
extra men. How shall the $580 be divided? 


426 


PARTNERSHIP 


5. X and Y engaged in business with a capital of $9000, of which 
X furnished f and Y At the end of 1 yr. each partner 
invested $4000 more. At the end of the second year they had 
resources to the amount of $13,000, and liabilities to the amount of 
$36,582. What was each partner’s net insolvency? 

6. Two men engaged in partnership for 2 yr. A invested at first 
$5000; 8 mo. later he put in $2000; and at the end of 18 mo. 
withdrew $1500. B invested at first $8000; at the end of 1 yr. he 
withdrew $1000; and 4 mo. later put in $3000. At the expiration 
of the two years their debts were all paid, and they had resources 
amounting to $21,675. Divide the gain or loss in proportion to 
average investment, and find each partner’s present worth. 

7. Three men engaged in partnership for 5 yr. from Jan. 1, 1920, 
on which date X made an investment of $6000, Y $8000, and 
Z $5000. July 1, 1921, X invested $4000 more; Y withdrew 
$2000, and Z put in $3000. Jan. 1, 1923, each partner increased 
his investment to $12,000. X was to have a salary of $1200; Y, of 
$1500; and Z, of $1800 per annum. If no salaries had been 
drawn and no profits divided until the expiration of the five years, 
what was each partner’s present worth at that time, their total re¬ 
sources being $87,500, and their liabilities $12,500? 

8. C and D began business July 1, 1923. C invested $10,000, 
and D invested $9000. Nov. 1, 1923, C withdrew $2000, and D 
invested $1000. Jan. 1, 1924, E was admitted to the partnership 
with an investment of $5000. Apr. 1, 1924, E invested $3000 
more, and D withdrew $1000. Dec. 31, 1924, D and E purchased 
C’s interest in the business. On that date their books showed the 
following resources and labilities: Cash, $18,750, merchandise, 
$17,000; bills receivable, $9600; accounts receivable, $8425; in¬ 
terest due on bills receivable, $237.50; real estate, $5000; accounts 
payable, $13,500; bills payable, $4800; interest on the same, 
$131.25; and sundry unpaid bills, $362.50. Good will was 
estimated at $3000, and $500 was reserved as a fund to meet bad 
debts. Each partner was entitled to interest at 6% on his net 
investment. Gains and losses were to be shared in the proportion 
of C f, D f, and E f. D and E were to pay C such amounts that 
their investments would be equal. How much should each pay C? 


BUILDING AND LOAN ASSOCIATIONS 


427 


BUILDING AND LOAN ASSOCIATIONS 

960. A building and loan association is a private corporation 
whose object is “to encourage industry, frugality, home building, 
and saving among its members.” To accomplish this object, each 
member of the association subscribes for one or more shares of stock 
(par value from $25 to $500, according to the laws of the various 
states), and agrees to pay for these in weekly or monthly install¬ 
ments of 25 £ (a week) or $1 (a month) per share. The dues so re¬ 
ceived are loaned to some member of the association who wishes to 
borrow the money for the purpose of buying or building a home. 
No one person is allowed to borrow more than the total par value of 
the shares he owns. He is charged interest at the legal rate. 

961. The sources of profit in an association are interest on loans, 
premiums on loans, fines against members who fail to pay their 
dues promptly, and the withdrawal of members who find it un¬ 
desirable to continue in the association. Persons withdrawing 
have returned to them all the dues they have paid on their shares, 
and a portion of the profits. The remaining portion of the profits 
is credited to shares still inforce. The book, or actual, value of 
a share before maturity is, therefore, somewhat greater than its 
withdrawal value. 

962. Building and loan associations are of three classes, viz., 
terminating, serial, and permanent. 

Shares in the terminating association are all dated alike, and 
hence all mature at the same time. A person entering the association 
after its organization is required to payback dues on his shares from 
their date. When the dues and profits together amount to the par 
value of the stock (generally $100 or $200), all shares are canceled, 
each member, if he has not borrowed from the' association, 
receives in cash the value of his shares, and the association ends. 

The serial plan of association is, in effect, a number of terminat¬ 
ing associations joined in one. That is to say, as many shares as 
possible dated Jan. 1, for instance, are sold. After a few meet¬ 
ings of the association there will be others who wish to join, but 
who do not wish to pay dues back to Jan. 1. Hence, a new 
series of shares is issued, in all respects like the first series, except 
that it is dated Apr. 1 or June 1, as the case may be, and ma- 


428 


BUILDING AND LOAN ASSOCIATIONS 


tures three or six months later than the first series. Additional 
series may be issued as occasion requires, thus making the associa¬ 
tion practically permanent. 

The permanent plan of association differs from others as follows: 

1. New members may join any time without paying back dues. 
Each member’s share or shares may be considered a series by itself. 

2 . Paid-up stock is issued. Shares of stock are not necessarily 
canceled when they mature. If the association is able to loan the 
funds, instead of canceling the stock, a certificate showing that the 
shares are fully paid is issued, and the holder thereof receives his 
semiannual dividends in cash. No more dues are paid on such 
shares, and they may be withdrawn at any time. 

3. Earnings are ascertained and divided semiannually, and when 
credited are subject to withdrawal the same as money payments. 

963. The serial plan is the most popular. The following prob¬ 
lems and discussion will be confined to that plan of association. 

964. To find the withdrawal value of a given number of shares. 

X owns 20 shares in a building and loan association and has 

paid dues at $1 per month per share for 6 yr. when he wishes to 
withdraw. If he is allowed profits at 4% per annum, to what 
amount is he entitled? 

72 X $20 = $1440, total amount paid in dues. 

36i 

$1440 X ~^2 X -04 = $175.20, profits. 

$1440 + $175.20 = $1615.20, withdrawal value. 

On 20 shares at $1 per month per share, the total amount of dues paid in is 
$1440. The first payment of dues ($20) has been earning profits for 72 mo., 
the second payment 71 mo., the third 70 mo., and so on to the 72d pay¬ 
ment, which has been earning profits only 1 mo. The investment is, there¬ 
fore, a decreasing arithmetical series having 72 as the first term and 1 as 
the last term, and the number of terms, 72. The sum of the series is equal to 
the product of the sum of the first and last terms multiplied by half the 
number of terms and equals, in this case, [(1 • + 72) X V]» or 2628. The 
usual method of computing the interest is to reckon it on the total dues paid 
in for the average, or equated, time. The average time is found by dividing 
the total number of months, 2628, by 72, the number of months dues have 
been paid, which gives 36f. The interest on $1440 for 36§ mo. at 4% is 
$175.20. The dues paid in, $1440, plus the profits, $175.20, gives $1615.20, 
the amount X can withdraw. 


BUILDING AND LOAN ASSOCIATIONS 


429 


PROBLEMS 

965. l. A has paid dues of SI a month per share on 18 shares 
in a series for 4 yr. He wishes to withdraw and is entitled to prof¬ 
its at 5% per annum. How much will he receive? 

2. B wishes to withdraw from his association after having paid 
his dues of SI a month per share on 25 shares regularly for 10 yr. 
If he is entitled to 6 % profits, what sum will he receive? 

3. C, after 5 yr., is unable to continue his payments of SI a 
month per share on 30 shares in a building and loan association, 
and withdraws therefrom. There are unpaid fines* against him 
amounting to S3.60. If his profits are estimated at 3J% per 
annum, what is the withdrawal value of his shares? 

966. To make distribution of profits. 

An association issued four series of shares as follows: First series, 
400 shares, dated Jan. 1, 1924; second series, 350 shares, dated 
July 1, 1924; third series, 500 shares, dated Jan. 1 , 1925; fourth 
series, 300 shares, dated July 1, 1925. The dues in each series 
were $1 per share per month. July 1, 1926, the entire profits 
were $2945.93. What was the value of a share of each series at 
that date? 

$30 X 400 X 15| = $186,000, first series’ investment for 1 month. 
$24 X 350 X 12 § = $105,000, second series’investment fori month. 
$18 X 500 X 9|= $85,500, third series’investment for 1 month. 
$12 X 300 X 6 | = $23,400, fourth series’investment for 1 month. 

$399,900, total investment for 1 month. 

Share of first series is ^IKHnnh or vltinr of $2945.93 = $1370.20. 
Share of second series is iinnHKS-, or of $2945.93 = $773.50. 
Share of third series is Sfe, or of $2945.93 = $629.85. 
Share of fourth series is or of $2945.93 = $172.38. 

$1370.20 -r- 400 = $3.43, profit on 1 share of first series. 

$773.50 -5- 350 = $ 2 . 21 , profit on 1 share of second series. 

$629.85 -r- 500 = $1.26, profit on 1 share of third series. 

$172.38 -T- 300 = $0.57, profit on 1 share of fourth series. 

$30 + $3.43 = $33.43, value of 1 share first series. 

•All unpaid fines are deducted from the amount due on shares. 







430 


BUILDING AND LOAN ASSOCIATIONS 


$24 + $2.21 = $26.21, value of 1 share second series. 

$18 + $1.26 = $19.26, value of 1 share third series. 

$12 + $0.57 = $12.57, value of 1 share fourth series. 

The principle that applies here is the same that applies to partnership settle¬ 
ment by average investment. Hence, first find the investment of each series 
for 1 mo. Dues of $1 a month on each of 400 shares have been paid for 30 mo. 
in the first series (the average time is 15§ mo. See explanation, page 428),. 
which makes an average investment of $30 X 400 X 15£ = $186,000 for 
1 mo. The average investments in the other series are found in the same 
way, making a total investment for 1 mo. of $399,900. Obviously, the share 
of the profits belonging to each series is in the ratio of their respective average 
investments, as shown in the solution. The profit per share in each series is 
found by dividing the entire profit belonging to each series by the number of 
shares in that series. The value of a share in each series is the sum of all dues 
paid on the share and the profit per share. 

PROBLEMS 

967. l. A building and loan association has issued a new series 
of shares at the beginning of each year. In the first series are 500 
shares; in the second series, 600 shares; in the third series, 400 
shares; and in the fourth series, 500 shares. Dues in all series are 
$1 per month. At the end of the fourth year, the profits are 
$5325. Find the value of a share in each series at that time. 

2. July 1, 1920, a building and loan association began business, 
and issued a first series of shares numbering 600. A second series 
of 500 shares was dated July 1, 1921. A third series of 700 shares 
was dated Jan. 1, 1923; and a fourth series of 800 shares was dated 
Jan. 1, 1924/ Dues in all series were $1 per month. Find the 
value of a share of each series, July 1, 1925, if the profits were 
$9575. 

968. In the association from which the following figures are taken a new 
series is open for subscriptions every 3 mo. Dues are 25 ^ a week per share. 
In this statement series No. 26 has been open 704 wk., there being 43 shares in 
the series, Dec. 31, 1923. The total subscription paid in on series 26 is equal 
to 25 i a week for 704 wk., or $176 per share, and on 43 shares, the total is 
$7568. During 1923, the subscriptions equal $559 (52 wk. at 25 i on each of 43 
shares). Deducting $559 from $7568 gives $7009, the total subscriptions paid 
in Dec. 31, 1922, which amount is earning profits during all of the year 1923. 
Profits for 1923 are paid on one half of the subscriptions paid in in 1923, or on 
$279.50, which added to $7009 gives $7288.50, the total amount on which 
profits are allowed for 1923 in series No. 26. The per cent of profit is found by 


SPEED TEST 


431 


dividing the total profit on all the series by the total subscriptions in all the 
series sharing in profits, as shown by the footing of the column headed “Total 
Profit-sharing Subscriptions, Dec. 31, 1923.” The same rate is paid on all 
series. 

969. Statement showing how Profits are Distributed 


Series No 

on 

M 

H 

£ 

GO 

W 

3 

3 

n 

cc 

Subscriptions in 
Full to Dec. 31, 
1923 

Subscriptions 
for 1923 

Subscriptions to 
Dec. 31, 1922 

Increase in 

Profit Sharing 

Subscription, 1923 

Total Profit 

Sharing Sub- 

scription, Dec. 31, 

1923 

Rate per cent 

of Profit 

Profit per Series 

26 

704 

43 

7,568 

559 

7,009 

279.50 

7,288.50 

61 

473.75 

27 

691 

9 

1,554.75 

117 

1,437.75 

58.50 

1,496.25 

61 

97.25 

28 

678 

68 

11,526 

884 

10,642 

442 

11,084 

61 

720.46 

29 

665 

27 

4,488.75 

351 

4,137.75 

175.50 

4,313.25 

61 

280.36 


PROBLEMS 


970. 1-4. Prepare a statement like the above for series 30, 31, 
32, 33, subscriptions at 25 i a week per share having been paid for 
652, 639, 626, 613, wk., respectively, the number of shares being 
36, 14, 45, 76, and the rate of profit being 7%. 

5 . A cooperative store, having a capital stock of 400 shares of 
$100 each, makes a profit on sales in one year of $12,500. The 
operating expenses ar $4800. Find the book value of 17 shares. 
If there is reserved for depreciation, reserve fund, etc., $2500, find 
the cash dividend that can be paid on 15 shares. 


EXAMINATIONS 


SPEED TEST 


971. Minimum time thirty minutes; maximum, one hour. De¬ 
duct one credit for each minute required beyond the minimum 
time. 

l. Find the market value in each of the following cases: 


Par Value 



Market Value 


$7,500 

7,900 

67,200 

475,200 
















432 


SPEED TEST 


2. Apportion the gains and losses according to investments in the 


following: 


A 

B 

$5,000 

$7,000 

3,000 

5,000 

2,500 

5,000 

15,000 

14,000 



$1,800 gain 

7,000 

7,500 gain. 

10,000 

7,000 loss. 

13,000 

840 loss. 


3. Equate the following account: 

June 1, Mdse., 60 da., $6000. July 1, Cash, $3000. 

16, Cash, 1500. 


4. Find the total duty on the following imports: 


Article 

Quantity 

Value 

Ad. Val. Duty 

Specific 

Total 

Duty 

Drugs 

2500 lb. 

£ 1000 

10 % 

If £ a pound 

_ 

Perfumery 

250 lb. 

F.3000 

50% 

60 i a pound 

— 

Rugs 

3000 sq. yd. 

£ 500 

40% 

10 i a square foot 

— 


5. Find cost of each, and the total cost of the following drafts: 


Face 

Rate Cost 

$1,500 

50 i premium per $1000 - 

1,875 

$1.00 discount “ “ - 

3,500 

$1.50 premium “ “ - 

5,000 

10 </t discount “ “ - 


Total .... 


6. Find the the total cost of the following exchange: 

£ 500 © 4.8520 - 

F. 2500 © 18.6 i - 

M. 4000© 22 £ - 

£ 1250 © 4.8665 - 

Total .... 

7. Find the par value of bonds necessary to produce income as 
follows: 


Bonds 

Income 

Par Value 

New York City 4f’s 

$1,800 

— 

United States 3’s 

2,400 

— 

C. B. & Q. 4’s 

2,000 

— 

Sea. Air Line 5’s 

3,000 

— 















WRITTEN TEST 


433 


8 . Add vertically, horizontally, and prove by finding the sum of 
the totals: 


$16,157 

48,287 

473,892 

$11,273 

2,380 

8,721 

$10,559 - 

16,272 - 

145,468 - 

9 . Find gain or loss and per 

10. Find the total interest on 

cent of gain or loss on cost: 

the following: 

Cost 

Selling Price 

$1200 for 50 days at 4£% 

$2.50 

$3.00 

1275 for 90 days at 6% 

3.60 

4.50 

2500 for 72 days at 5%. 

5.00 

120.00 

6.50 

110.40 

3600 for 13 days at 7%. 


WRITTEN TEST 

972. l. A sight draft on Chicago for $1771 is purchased at |% 
premium. Find the cost. 


2 . Divide $5000 into four parts proportional to 8.407, 5.149, 
4.132, and 2.312. 


3 . A lot of woolen cloth weighing 1200 lb. is invoiced in Liver¬ 
pool as 1575 yd. at 9s. 6 d. per yd. Find the amount of duty 
paid in New York at 30 i a pound and 45% ad valorem. 

• 

4 . At 4.8725, what is the cost of a bill of exchange to settle for 
the cloth in problem 3? At what price per yard must it be sold to 
gain 25%? 

5 . My broker buys for me 200 shares C. M. & St. P. at 18J, 
brokerage J%. Thirty days later, I instruct him to sell at 19 f. 
Allowing \% for selling, find my gain or loss. 

6 . If the cost of erecting a $28,000 building is assessed on 
$450,000 capital stock, how much must a stockholder pay who 
owns 100 shares at $100 each? 

7 . A and B are in partnership. Gains and losses are shared in 
the ratio of J to A and J to B. June 30, 1924, their stock is worth 
$6000. Each is entitled to 5% interest on his investment from 
Jan. 1, 1924. Find from the following trial balance each partner’s 
present worth, June 30, 1924. 

VAN TUYL’S NEW COMP. AR —28 



434 


WRITTEN TEST 


Trial Balance June 30, 1924 


A, Capital . 



$10,900 

A, Drawings, including interest . . . 

$900 



B, Capital . 



5,500 

B, Drawings, including interest . . . 

255 



Accounts Payable. 



7,419 

Sales. 



53,820 

Stock, Jan. 1, 1924 . 

6,014 



Accounts Receivable (all good) .... 

16,997 



Cash, in hand. 

3,489 



Bad debts. 

195 



Cartage. 

1,019 



Discount. 

2,509 



Fire insurance. 

67 



Wages . 

4,553 



Rent, taxes. 

3,244 



Repairs. 

202 



Salaries. 

1,624 



Office Expenses . 

271 



Purchases. 

36,300 




8 . On the Panama Canal Bonds, interest is payable quarterly— 
on the 1st of February, May, August, and November, the annual 
rate being 2%. Find the cost, including interest, of $50,000 of 
these bonds on the 10th of June at 99f. 


9 . Find the equated date: 

1924 1924 


Mar. 

1, 

Mdse., 2 mo. 

$1500 

Apr. 1, 

Cash, 

$500 

Apr. 

10, 

” 1 mo. 

750 

30, 

a 

1000 

May 

1, 

” Net, 

1000 

May 1, 

Note, 2 mo. 

1500 

10. 

1924 

Find 

the balance due Jan. 1 

, 1925. 

1924 

Interest 6%. 


Oct. 

1, 

Balance, 

$1546.72 

Oct. 10, 

Cash, 

$1200 


30, 

Mdse., 2 mo. 

1268.25 

Nov. 15, 

Note, 1 mo. 

1000 

Dec. 

15, 

” net, 

1143.88 

Dec. 10, 

Note, 2 mo. 
with int. 

1200 


WRITTEN TEST 

New York State Regents’ Examination Questions 

973. 1. Find the cost of 3 tons 430 lb. of coal at $12.75; and 15 
pieces of hemlock 6" X 8" X 20' at $55 per M. 

























WRITTEN TEST 


435 


2 . How many yards of carpet 27 in. wide will be required to 
carpet the floor of a room 24 ft. long and 18 ft. wide, if the strips 
run the long way of the room, and 6 in. are allowed on each strip 
for loss in matching? 

3 . What must be the list price of goods that cost $410 in order 
to make a profit of 20 % of cost, if they are to be sold at a discount 
of 20 % and 10 % from the list price? 

4 . On May 2, 1924, J. Williams bought the following goods less 
a discount of 5%, terms 2/10, n/60; 5 pieces velveteen, 42, 44,43, 
42,45 yd. at 25 ^ a yard. Williams sold the goods May 3 at 35 i a 
yard, terms 1/10, n/30. He received cash for his sale on May 9, 
and paid for his purchase on May 10. Find his profit. 

5 . On an asking price of $400, which is the more favorable for 
the buyer, and how much: a discount of 30%, 20%, and 12§%, or 
a discount of 40%, 12J%, and 10%? 

6. On July 1, 1924, a trader's stock of goods was destroyed by 
fire, but he saved his books of record; his goods were fully insured, 
and he proved his loss from the following facts: value of goods on 
hand Jan. 1, 1924, $21,500; purchases from Jan. 1,1924 to July 1, 
$54,300; sales from Jan. 1 to July 1 , $63,750. His records showed 
that all sales were made at an average advance of 25% above cost. 
Find the value of the stock destroyed by fire. 

7 . A grain merchant received $22.26 for selling a quantity of 
grain; if he charged 2 % commission and sold the grain at $1 06 
per bushel, how many bushels did he sell? 

8 . A 4 months' note for $875, without interest, dated May 17, 
1924, was discounted at a bank July 11, 1924, at 6 %. Find the 
discount and the proceeds. 

9 . On a note for $1200 dated May 17, 1924, payable on demand 
with interest at 6 %, a payment of $500 was made Dec. 8 , 1924. 
How much was due Feb. 16, 1925? 

10 . A city made an appropriation of $53,040 for its public 
schools. The assessed value of real estate was $6,710,000 and of 
personal property $2,130,000. What was the tax rate for school 
purposes? How much was Brown’s school tax if his property was 
assessed at $12,500? 


436 


WRITTEN TEST 


WRITTEN TEST 

Questions from Business Educators’ Association, Canada 
974. l. Find the cost of a carpet f yard wide, at $1.25 per yard, 
for a room 22 ft. X 18 ft. if the strips run lengthwise and there is a 
waste of i of a yard on each strip for matching the pattern. 

2. An article marked 62J% above cost was sold less 25% and 
20%. If a collector was afterwards paid 20% for collecting the 
account, what was the gain or loss per cent? 

3. Having bought a house of A at 12J% less than it cost him 
I spent $430 for repairs and sold it for $7293, thereby gaining 10% 
on my investment. How much did the house cost A? 

4. Equate the following account: 


1924 

Aug. 10, 

To Mdse., 30 da. 

$375 

1924 

Sept. 3, 

By Draft, 30 da. 

$250 

Aug. 25, 

To Cash, 

450 

Sept. 27, 

By Cash, 

425 

Sept. 10, 

To Mdse., 2 mo. 

175 

Oct. 1, 

By Cash, 

25 


5. I sold 2978 bu. wheat at $1.05 a bushel, and invested the 
proceeds in sugar, reserving my commission of 5% for selling and 
1|% for buying, and the expenses of shipping, $53.57. How 
much did I invest in sugar? 

6. A man invests $12,000 in 3% stock at 75. He sells out at 
90, and invests | of the proceeds in 3J% stock at 80 and the re¬ 
mainder in 6% stock at 128. Find the change in his income. 

7. I bought two houses for $11,700, paying 25% more for one 
than for the other. I sold the cheaper house at 20% profit and the 
higher priced one at 16f % profit. What was my total gain? 

8. A stock of goods was marked at 22^% advance on cost, but 
becoming damaged was sold at 20% discount on the marked price, 
which caused a loss of $1186.40. Find the cost of the goods. 

9. A note of $1500 dated June 20, 1923, bearing interest at 5%, 
had payments indorsed as follows: Dec. 5, 1923, $300; April 2, 
1924, $10; July 20, 1924, $500; Dec. 31, 1924, $400. Find the 
amount due June 22, 1925. 

10. An invoice of woolen cloth, imported from England, was 
valued at £956 6s. If its weight was 684 lb., how much was the 
duty at 50 cents per pound specific, and 35% ad valorem? 


INDEX 


Abstract purchase book, 56. 
Abstract sales book, 48. 
Acceptance, 316. 

Account purchase, 292. 

Account sales, 292. 

Accounts current, 419. 

Accurate interest, 313. 

Addends, 38. 

Addition, 38. 
checking by elevens, 42. 
checking by nines, 41. 
of denominate numbers, 158. 
of fractions, 88. 
principles of, 116. 

Aliquant parts (note), 12. 

Aliquot parts, 12. 
application of, to division, 29. 
application of, to simple in¬ 
terest, 32. 

application of, to trade dis¬ 
count, 22. 
eighths, 13. 

halves and quarters, 12. 
of 60, 32. 

sevenths, ninths, and fif¬ 
teenths, 19. 
sixteenths, 14. 
thirds and sixths, 16. 
twelfths, 16. 

Alloy in United States coins, 11. 
Altitude, 169, 180. 

Amount, 230. 

Angle, 167. 

Angular measure, 144. 

Antecedent, 128, 129. 
Apothecaries fluid measure, 140. 
Application of the fundamental 
principles of arithmetic, 117. 
Approximate measures, 139, 213. 
Arabic notation, 8. 

Area, of parallelogram, 169. 
of circle, 176. 
of triangle, 171. 

Assessment, 373. 

Assessment roll, 344. 

Assessors, 344. 


Assignment, 371. 

Atchison, Topeka and Santa Fe 
Time Table, 196. 

Average, 127. 

Average investment, 422. 
Avoirdupois weight, 142. 

Bank discount. 320. 

Bank draft, 315. 

Bankers’ 60-day method, 303. 
Bankers’ time, 195. 

Bankers’ time table, 324. 

Base, 230. 

Base, of plane figure, 169. 

Bear, 373. 

Beneficiary, 360. 

Bill of exchange, 395. 

Billing and trade discount, 250. 
Bin, 213. 

Blank indorsement, 319. 

Board foot, 209. 

Bond, 381. 

compared with stocks, 383. 
coupon,383. 
illustration of, 382. 
quotations, 384. 
registered, 383. 
table, 387. 

Bonded warehouse, 407. 

Book paper, 206. 

Breakage, 406. 

Broker, 373. 

Brokerage, 292. 

Building and Loan Association, 
427. 

earnings, 428. 
permanent plan, 428. 
serial plan, 427. 
terminating plan, 427. 

Bull, 374. 

Bundle of shingles, 204. 

Cable transfer, 398. 

Cancellation, 74. 

Capacity 213. 

Capital. 421 


Capital stock, 370. 

Carat, 142. 

Carpeting, 198. 

Cash balance, 419. 

Cash discounts, 251. 

Casting out elevens, 42, 64, 67. 
Casting out nines, 41, 64, 67. 
Cause and effect, 130. 

Certificate of deposit, 319. 
Charter, 370. 

Check, 314. 

Checking the work, 41, 52, 63, 67. 
Circle, 169. 

Circular measure, 144. 
Circumference, 169. 

Cisterns and reservoirs, 214. 

Coal, 213. 

Coins of United States, 11. 
Collaterals, 374. 

Collector, 344. 

Collector of customs, 407. 
Commercial draft, 315. 
Commission and brokerage, 292 
Commission merchant, 292. 
Common stock, 370. 

Commutative law of multiplica¬ 
tion, 117. 

Comparison, of measures, 154. 

of weights, 143. 

Complement method, 54. 
Complement of a number, 54. 
Compound interest, 332. 

tables, 332, 333, 336. 

Compound proportion, 131. 
Compound time, 194. 

Cone, 180. 

Consequent, 128, 129. 

Consignee, 292. 

Consignment, 292. 

Consignor, 292. 

Consumption entry, 407. 

illustration of,'408. 

Conversion table, 404. 
Corporation, 370. 

Cost by the 100,1000, and ton, 29. 
Cost of doing business, 272 


437 




438 


INDEX 


Counting table, 148. 

Coupon, 383. 

illustration, 383. 

Coupon bond, 383. 

Cube, 180. 

Cubic measure, 139. 
metric, 150. 

Currency memorandum, 104. 
Custom house, 407. 

Customs, 406. 

Cylinder, 180. 

Date of maturity (notes), 1, 2, 3, 
4, 321. 

Days of grace, 321. 

Decimals, 80. 

Denominate numbers, 138. 
compound, 138. 
reduction ascending, 153. 
reduction descending, 152. 
simple, 138. 
tables, of, 138 148. 

Diagonal, 168. 

Diameter, 169. 

Difference, 230. 

Direct tax, 343. 

Discount series, 258. 

Dividends, 373. 

Divisibility, tests of, 69. 

Division, 65. 

by continued subtraction, 68. 
checking, 67. 

of denominate numbers, 161. 
of fractions, 108. 

Domestic exchange, 390. 

Dormant partner, 421. 

Drawee, 315. 

Drawer, 315. 

Drill exercise, addition, 49. 

aliquot parts, 28. interest, 37. 
percentage, 242, profit and loss, 
278 

Dry measure, 141. 

Duties, ad valorem, 406. 
specific, 406. 

Elevens, casting out, 42, 64, 67. 
Endowment policy, 360. 

English consols, 381. 

English money, 146. 

Equated date, 413. 

Equation of accounts, 412, 415. 
Equation of payments, 412 
counting back, 417. 
counting forward, 418. 
Equilateral triangle, 168. 


Estimating, 218. 

Evolution, 163. 

Exact time, 194. 

Examinations, 134, 227, 299, 367, 
431. 

Exchange, 390. 
centers, 391. 
domestic, 390. 
foreign, 395. 

Explanation of stock exchange 
terms, 373. 

Exponent, 162. 

Express money order, 391. 

Extremes, 129. 

Factor, 57, 69. 

Factoring, 69. 

Factory expense, 284. 

Farm problems, 219. 

Fire insurance, 353. 

Flat paper, 206. 

Flooring, 203. 

Floor plan, 202. 

Focal date, 413. 

Food products, 248. 

Foreign coins, value of, 399. 

Foreign exchange, 395. 

Fractional equivalents, 232. 

Fractions, 79. 
addition, 88. 
dissimilar, 87. 
division, 108. 
improper, 83. 
interchange of forms, 85. 
lowest terms of, 81. 
multiplication, 93 
proper, 84. 
reduction of, 81. 
similar, 86. 
subtraction, 91. 
terms, 81. 

Franc, 147. 

Free list, 408. 

French money, 147. 

French rentes, 383. 

Fundamental principles of arith¬ 
metic, 116. 
application of, 117. 

General partner, 421. 

German money, 147 

Gold coins of the United States, 11 
fineness and weight, 11. 

Gold, the standard of monetary 
value, 397. 
ratio to silver, 11. 


Graphs, 221. 

Greatest common divisor, 71. 

by inspection, 71. 

Gross cost, 268, 292. 

Gross proceeds, 292. 

Gross profit, 268. 

Guaranty, 292. 

Horizontal addition, 45. 
Horizontal line, 167. 

How to read numbers, 8. 
Hundreds, to find cost by, 29. 
Hypotenuse, 168, 173. 

Import duties, 409. 

Improper fraction, 83. 

Income tax, 351. 

Indirect tax, 343. 

Indorsement, 319. 

kinds of, 319. 

Insolvency, 422. 

Insurance, 353.. 
fire, 353. 
life, 360. 

Insurance company, 353. 
Interchanging principal and time, 
309. 

Interest, accurate, 313. 
bankers’ 60-day method, 303. 
by application of aliquot parts 

32. 

compound, 332. 
simple, 32, 302. 

Interest term, 364. 

Invoices, 250 252. 

Involution, 162. 

Isosceles triangle, 168. 

Karat, 142. 

Knot, 139. 

Land, 216. 

Land measure (metric), 149. 
Leakage, 406. 

Leap year, 145. 

Least common denominator, 86. 
Least common multiple, 72. 

by inspection, 73. 

Legal rate of interest, 303. 

Letter of credit, 398. 

illustration, 398. 

Liabilities, 421. 

License fee, 343. 

Life insurance, 360. 
table of annual premiums, 362. 
table of values, 361. 






INDEX 


439 


Limited partner, 421. 

Line, 167. 

Linear measure, 138. 

Linear metric measure, 149. 

Liquid and dry measures, com¬ 
pared, 141. 

Liquid measure, 140. 

“Long” (term in stocks), 374. 

Long ton,143, 407. 

Loss by fire, 353. 

Lumber, 209. 

Lumber yard practice, 210. 

Manufacturing costs, profits, etc., 
284. 

Marking goods, 287. 

Means, 129. 

Measure, 138. 
approximate, 139. 
standard unit of, 138. 

Measures of capacity, 140. 
apothecaries’ fluid, 140. 
dry, 141. 
liquid, 140. 
metric, 150. 

Mensuration, 167. 

Merchants’ rule, 341. 

Meter, 148. 

Metric equivalents, 151. 

Metric system, 148. 

Miscellaneous measures, 147. 

Mixed numbers, 84. 

Model invoice, 250. 

Model statement, 263. 

Money, reduction of, 155. 

Multiple, defined, 72. 

Multiplication, 57. 
checking, 63. 
commutative law of, 117. 
of denominate numbers, 160. 
of fractions, 93. 
suggestions for shortening, 23. 

Negotiable paper, 314. 
bank draft, 315. 
certificate of deposit, 319. 
check, 314. 
promissory note, 318. 
sight draft, 316. 
time draft (after date), 315. 
time draft (after sight), 316. 
trade acceptance, 317. 

Net cost rate factor, 259. 

Net gain, 422. 

Net investment, 421. 

Net loss, 422. 


Net proceeds, 292. 

Net profit, 268. 

Net sales, 268. 

Newspaper, 206. 

New York Standard (80%) average 
clause, 353. 

Nines, casting out, 41,64, 67. 
Nominal partner, 421. 

Notation, Arabic, 8. 

Roman, 7. 

Notice of protest, 320. 

Open policy, 354. 

Open policy book, 354. 

Ordinary interest, 303. 

Paid up policy, 361. 

Paper and books, 206. 

Paper table, 147. 

Parallel lines, 167. 

Parallelogram, 147. 

Par of exchange, 397. 

Parcel post, 76. 

Partial payments, 339. • 
by merchants’ rule, 341. 
by United States rule, 339. 
Partitive proportion, 132. 
Partners, 421. 
dormant or secret, 421. 
general, 421. 
nominal, 421. 
special or limited, 421. 
Partnership, 421. 

Par value, 370. 

Payee, 315. 

Pay rolls, 103. 

differential rate plan, 107. 

Pay roll check, 105. 

Percentage, 229. 

drill, 242. 

Perimeter, 169. 

Perpendicular line, 167. 

Pitch of roofs, 204. 

Policy of insurance, 353. 
open, 354. 
valued, 354. 

Poll tax, 343. 

Polygon, 169. 

Port of delivery, 407. 

Port of entry, 407. 

Postal money order, 391. 

Postal rates, 76. 

Postal savings banks, 366. 

Power of a number, 162. 

Practical measurements, 194. 
bankers’ time, 195. 


capacity, 213. 

carpeting, 198. 

compound time, 194. 

counting forward or back, 197. 

exact time, 194. 

flooring, 203. 

land, 216. 

lumber, 209. 

lumber yard practice, 210. 
painting, 198. 
paper and books, 206. 
papering, 198. 
paving, 205. 
plastering, 198. 
roofing, 203. 

Preferred stock, 370. 

Premium, 353, 360. 
insurance table of, 362. 

Present worth and true discount 
328. 

Present worth, of a debt, 328. 
of a fir^n, 421. 

Prime cost, 268, 284. 

Prime factor, 69. 

Prime number, 69. 

Principal, 32, 292, 302. 

Principal meridian, 216. 

Principles of arithmetic, 116. 

Prism 179. 

Proceeds, of a note, 320. 

Profit, 268. 

Profit and loss, 268. 
drill chart, 278. 

Promissory note, 318. 

Proper fraction, 84. 

Property tax, 343. 

Proportion, 129. 
compound, 131. 
partitive, 132. 
principles of, 130. 
simple, 130. 

Protest, 320. 

Public store, 407. 

Pyramid, 180. 

Quadrant, 144. 

Quadrilateral, 167. 

Qualified indorsement, 319. 

Radical sign, 163. 

Radius, 169. 

Ranges, 216. 

Rate, 230. 

Rate of interest, 302. 

Rates of exchange, 400. 





440 


INDEX 


Ratio, 128. 

direct and inverse, 128. 
sign of, 128. 

simple and compound, 128. 
terms, 128. 

Reading and writing numbers, 7-9. 
Reciprocal of a fraction, 111. 
Rectangle, 168. 

Reduction of denominate num¬ 
bers, 152. 

ascending, descending, 152, 153. 
Registered bond, 383. 

Reservoirs, 214. 

Resources, 421. 

Right angle, 167. 

Right-angled triangle, 168. 

Roman notation, 7. 

Roofing, 203. 

Root, 163. 

Savings banks, 364. 

Schedule of amortization', 337. 
Secret partner, 421. 

Section of land, 217. 

Set of exchange, 396. 

Sextant, 144. 

Shipment, 292. 

Short, 374. 

Sight draft, 316. 

Sign, 144. 

Silver coins of the United States, 11. 

ratio to gold, 11. 

Similar solids, 190. 

Similar surfaces, 188. 

Simple interest, 32, 302. 

Sinking funds, 335. 

table, 336. 

Sizes of books, 207. 

Sizes of paper, 206. 

Slant height, 180. 

Solids, 179. 

Special indorsement, 319. 

Special partner, 421. 

Specific gravity, 191. 
table of, 192. 

Speed tests, 134, 135,227,299,367, 
431. 

Square, 139, 168. 

Square measure, 139. 

metric, 149. 

Square root, 163. 

Standard short rate scale, 357. 
Standard time, 145. 

map of belts, 146. 

Standard unit of measure, 138. 


Standard unit of weight, 141. 
Statement, 263. 

Stock broker, 373. 

Stock certificate, 370. 

form of, 371. 

Stockholder, 370. 

Stocks, 370. 

commission for buying or selling, 
373. 

market quotations, 372. 
tax on sales of, 373. 

Subtraction, 50. 
checking the work, 52. 
complement method of, 54. 
making change or Austrian 
method of, 52. 

of denominate numbers, 159. 
of fractions, 91. 

Summary of principles of interest, 
308. 

Surfaces, 167. 

Tables, apothecaries fluid measure, 
140. 

apothecaries weight, 142. 
avoirdupois weight, 142. 
bond, 387. 

circular measure, 144. 
conversion, 404. 
counting, 148. 
cubic measure, 139. 
denominate numbers, 138-148. 
dry measure, 141. 

English money, 146. 
fractional equivalents, 232. 
French money, 147. 

German money, 147. 
import duties, 409. 
insurance premiums, 362. 
linear measure, 138. 
liquid measure, 140. 
long ton, 143. 
metrical equivalents, 151. 
net cost rate factors, 259. 
paper, 147. 
short rate scale, 357. 
square measure, 139. 
time, 144. 

Troy weight, 141. 
values, life insurance, 361. 

Tare, 406. 

Tariff, 408. 

Tariff Act, 1923, 409. 

Tax bill, 349. 

Taxes, how apportioned, 343. 
Telegraph money order, 392. 


Term of discount, 320. 

and note 1, 321. 

Term of insurance, 353. 

Term policy, 360. 

Tests of divisibility, 69. 
Thousands, to find the cost by, 29- 
Time, 144. 

Time card, 106. 

Time draft, after date, 315. 

after sight, 316. 

Time sheet, 105. 

Time table, Atchison, Topeka and 
Sante Fe Railroad, 196. 

Toll charges for telegrams, 393. 
Ton, to find cost by, 29. 
Township, 216. 

Trade discount, 22, 250. 

Travelers’ check, 400. 

illustrated, 401. 

Triangle, 168. 

Troy weight, 141. 

True discount, 328. 

Type problems, 118, 230. 

United States customs, 406 
United States money, 10. 
kinds of, 10. 

ratio of gold to silver, 11. 

United States rule, 339. 

Usury, 303. 

Value of foreign coins, 399. 

Valued policy, 354. 

Vertical line, 167. 

Volume, of cone, 184. 
cylinder, 180. 
prism, 180. 
pyramid, 184. 

Waiver clause, 354. 

Warehouse entry, 407. 

Weight, 141. 
apothecaries, 142. 
avoirdupois, 142. 
comparison of, 143. 
long ton, 142. 
metric, 150. 
of coins, 11. 

Troy, 141. 

Whole life policies, 360. 

Wood, 212. 

Wood measure, metric, 150. 
Wrapping paper, 206. 

Written tests, 136, 228, 300, 368, 
433-436. : 

Zones, postal, 77. 














































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